Commutativity in Double Interchange Semigroups Murray Bremner Joint work with Gary Au, Fatemeh Bagherzadeh, and Sara Madariaga Department of Mathematics and Statistics, University of Saskatchewan Workshop on Operads and Higher Structures in Algebraic Topology and Category Theory July 29 – August 2, 2019, University of Ottawa w x y z = w x y z = w x y z Diagrammatic representation of the interchange law 1 / 56
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Transcript
Commutativity in Double Interchange Semigroups
Murray BremnerJoint work with Gary Au Fatemeh Bagherzadeh and Sara Madariaga
Department of Mathematics and Statistics University of Saskatchewan
Workshop on Operads and Higher Structures inAlgebraic Topology and Category Theory
July 29 ndash August 2 2019 University of Ottawa
w x
y z=
w x
y z=
w x
y z
Diagrammatic representation of the interchange law
1 56
A very gentle introduction tohigher-dimensional algebra
2 56
Horizontal and vertical multiplications
Suppose that x and y are two indeterminates
When we compose x and y we usually write the result as xy
This composition may not be commutative xy 6= yx in general
We write the products xy and yx horizontally mdash but why
Why donrsquot we write the products vertically yx or x
y
Paper (or blackboards or computer screens or ) are 2-dimensional
Why donrsquot we use the third dimension (orthogonal to the screen)
We could write x in front of y or y in front of x using 3 dimensions
Is this just silly or could it possibly have some important applications
Multiplying in different directions leads to higher-dimensional algebra
3 56
From Ronald Brownrsquos survey paper Out of Line
4 56
A crossword puzzle is this higher-dimensional algebra
5 56
The interchange law for two operations
Consider the 2times 2 square product y zw x of four variables
There are two different ways to write this using and bull
(w x) bull (y z) or (w bull y) (x bull z)
Note the transpositions of x and y and of the operations and bull
We assume these two ways of writing y zw x always give the same result
This assumption is called the interchange law for all w x y z we have
(w x) bull (y z) = (w bull y) (x bull z)
This relation between the two operations can be expressed as
w x
y z=
w x
y z=
w x
y z
6 56
The surprising Eckmann-Hilton argument (1)
Assume that and bull satisfy the interchange law
(w x) bull (y z) = (w bull y) (x bull z)
Assume that and bull have identity elements 1 and 1bull respectively
In the interchange law take w = z = 1 and x = y = 1bull
(lowast) (1 1bull) bull (1bull 1) = (1 bull 1bull) (1bull bull 1)
By definition of identity elements
1 1bull = 1bull 1 = 1bull 1 bull 1bull = 1bull bull 1 = 1
Using these to simplify equation (lowast) gives 1bull bull 1bull = 1 1
By definition of identity elements this implies that 1 = 1bull
The two identity elements are equal
7 56
The surprising Eckmann-Hilton argument (2)
Write 1 for the common identity element 1 = 1 = 1bull
In the interchange law take w = z = 1
(1 x) bull (y 1) = (1 bull y) (x bull 1)
By definition of identity element this implies that x bull y = y x
The black operation is the opposite of the white operation
Using this to simplify the interchange law gives
(y z) (w x) = (y w) (z x)
Taking x = y = 1 in the last equation gives
(1 z) (w 1) = (1 w) (z 1)
By definition of identity element this implies that z w = w z
The white operation is commutative
8 56
The surprising Eckmann-Hilton argument (3)
But if x bull y = y x and y x = x y then x bull y = x y
The two operations are equal (There is really only one operation)
Hence the interchange law can be simplified to
(w x) (y z) = (w y) (x z)
Taking x = 1 gives
(w 1) (y z) = (w y) (1 z)
By definition of identity element this implies
w (y z) = (w y) z
The operation is associative
We have proved the (in)famous Eckmann-Hilton Theorem
9 56
The Eckmann-Hilton Theorem
Theorem (Eckmann-Hilton)
Let S be a set with two binary operations S times S rarr S denoted and bullSuppose that these two operations satisfy the interchange lawSuppose also that these two operations have identity elements 1 and 1bullThen the two identity elements are equal the two operations are equaland the single remaining operation is both commutative and associative
Beno Eckmann Peter HiltonGroup-like structures in general categories IMultiplications and comultiplicationsMathematische Annalen 145 (196162) 227ndash255
bull Theorem 333 (page 236)
bull The definition of H-structure (page 241)
bull Theorem 417 (page 244)
10 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
A very gentle introduction tohigher-dimensional algebra
2 56
Horizontal and vertical multiplications
Suppose that x and y are two indeterminates
When we compose x and y we usually write the result as xy
This composition may not be commutative xy 6= yx in general
We write the products xy and yx horizontally mdash but why
Why donrsquot we write the products vertically yx or x
y
Paper (or blackboards or computer screens or ) are 2-dimensional
Why donrsquot we use the third dimension (orthogonal to the screen)
We could write x in front of y or y in front of x using 3 dimensions
Is this just silly or could it possibly have some important applications
Multiplying in different directions leads to higher-dimensional algebra
3 56
From Ronald Brownrsquos survey paper Out of Line
4 56
A crossword puzzle is this higher-dimensional algebra
5 56
The interchange law for two operations
Consider the 2times 2 square product y zw x of four variables
There are two different ways to write this using and bull
(w x) bull (y z) or (w bull y) (x bull z)
Note the transpositions of x and y and of the operations and bull
We assume these two ways of writing y zw x always give the same result
This assumption is called the interchange law for all w x y z we have
(w x) bull (y z) = (w bull y) (x bull z)
This relation between the two operations can be expressed as
w x
y z=
w x
y z=
w x
y z
6 56
The surprising Eckmann-Hilton argument (1)
Assume that and bull satisfy the interchange law
(w x) bull (y z) = (w bull y) (x bull z)
Assume that and bull have identity elements 1 and 1bull respectively
In the interchange law take w = z = 1 and x = y = 1bull
(lowast) (1 1bull) bull (1bull 1) = (1 bull 1bull) (1bull bull 1)
By definition of identity elements
1 1bull = 1bull 1 = 1bull 1 bull 1bull = 1bull bull 1 = 1
Using these to simplify equation (lowast) gives 1bull bull 1bull = 1 1
By definition of identity elements this implies that 1 = 1bull
The two identity elements are equal
7 56
The surprising Eckmann-Hilton argument (2)
Write 1 for the common identity element 1 = 1 = 1bull
In the interchange law take w = z = 1
(1 x) bull (y 1) = (1 bull y) (x bull 1)
By definition of identity element this implies that x bull y = y x
The black operation is the opposite of the white operation
Using this to simplify the interchange law gives
(y z) (w x) = (y w) (z x)
Taking x = y = 1 in the last equation gives
(1 z) (w 1) = (1 w) (z 1)
By definition of identity element this implies that z w = w z
The white operation is commutative
8 56
The surprising Eckmann-Hilton argument (3)
But if x bull y = y x and y x = x y then x bull y = x y
The two operations are equal (There is really only one operation)
Hence the interchange law can be simplified to
(w x) (y z) = (w y) (x z)
Taking x = 1 gives
(w 1) (y z) = (w y) (1 z)
By definition of identity element this implies
w (y z) = (w y) z
The operation is associative
We have proved the (in)famous Eckmann-Hilton Theorem
9 56
The Eckmann-Hilton Theorem
Theorem (Eckmann-Hilton)
Let S be a set with two binary operations S times S rarr S denoted and bullSuppose that these two operations satisfy the interchange lawSuppose also that these two operations have identity elements 1 and 1bullThen the two identity elements are equal the two operations are equaland the single remaining operation is both commutative and associative
Beno Eckmann Peter HiltonGroup-like structures in general categories IMultiplications and comultiplicationsMathematische Annalen 145 (196162) 227ndash255
bull Theorem 333 (page 236)
bull The definition of H-structure (page 241)
bull Theorem 417 (page 244)
10 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Horizontal and vertical multiplications
Suppose that x and y are two indeterminates
When we compose x and y we usually write the result as xy
This composition may not be commutative xy 6= yx in general
We write the products xy and yx horizontally mdash but why
Why donrsquot we write the products vertically yx or x
y
Paper (or blackboards or computer screens or ) are 2-dimensional
Why donrsquot we use the third dimension (orthogonal to the screen)
We could write x in front of y or y in front of x using 3 dimensions
Is this just silly or could it possibly have some important applications
Multiplying in different directions leads to higher-dimensional algebra
3 56
From Ronald Brownrsquos survey paper Out of Line
4 56
A crossword puzzle is this higher-dimensional algebra
5 56
The interchange law for two operations
Consider the 2times 2 square product y zw x of four variables
There are two different ways to write this using and bull
(w x) bull (y z) or (w bull y) (x bull z)
Note the transpositions of x and y and of the operations and bull
We assume these two ways of writing y zw x always give the same result
This assumption is called the interchange law for all w x y z we have
(w x) bull (y z) = (w bull y) (x bull z)
This relation between the two operations can be expressed as
w x
y z=
w x
y z=
w x
y z
6 56
The surprising Eckmann-Hilton argument (1)
Assume that and bull satisfy the interchange law
(w x) bull (y z) = (w bull y) (x bull z)
Assume that and bull have identity elements 1 and 1bull respectively
In the interchange law take w = z = 1 and x = y = 1bull
(lowast) (1 1bull) bull (1bull 1) = (1 bull 1bull) (1bull bull 1)
By definition of identity elements
1 1bull = 1bull 1 = 1bull 1 bull 1bull = 1bull bull 1 = 1
Using these to simplify equation (lowast) gives 1bull bull 1bull = 1 1
By definition of identity elements this implies that 1 = 1bull
The two identity elements are equal
7 56
The surprising Eckmann-Hilton argument (2)
Write 1 for the common identity element 1 = 1 = 1bull
In the interchange law take w = z = 1
(1 x) bull (y 1) = (1 bull y) (x bull 1)
By definition of identity element this implies that x bull y = y x
The black operation is the opposite of the white operation
Using this to simplify the interchange law gives
(y z) (w x) = (y w) (z x)
Taking x = y = 1 in the last equation gives
(1 z) (w 1) = (1 w) (z 1)
By definition of identity element this implies that z w = w z
The white operation is commutative
8 56
The surprising Eckmann-Hilton argument (3)
But if x bull y = y x and y x = x y then x bull y = x y
The two operations are equal (There is really only one operation)
Hence the interchange law can be simplified to
(w x) (y z) = (w y) (x z)
Taking x = 1 gives
(w 1) (y z) = (w y) (1 z)
By definition of identity element this implies
w (y z) = (w y) z
The operation is associative
We have proved the (in)famous Eckmann-Hilton Theorem
9 56
The Eckmann-Hilton Theorem
Theorem (Eckmann-Hilton)
Let S be a set with two binary operations S times S rarr S denoted and bullSuppose that these two operations satisfy the interchange lawSuppose also that these two operations have identity elements 1 and 1bullThen the two identity elements are equal the two operations are equaland the single remaining operation is both commutative and associative
Beno Eckmann Peter HiltonGroup-like structures in general categories IMultiplications and comultiplicationsMathematische Annalen 145 (196162) 227ndash255
bull Theorem 333 (page 236)
bull The definition of H-structure (page 241)
bull Theorem 417 (page 244)
10 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
From Ronald Brownrsquos survey paper Out of Line
4 56
A crossword puzzle is this higher-dimensional algebra
5 56
The interchange law for two operations
Consider the 2times 2 square product y zw x of four variables
There are two different ways to write this using and bull
(w x) bull (y z) or (w bull y) (x bull z)
Note the transpositions of x and y and of the operations and bull
We assume these two ways of writing y zw x always give the same result
This assumption is called the interchange law for all w x y z we have
(w x) bull (y z) = (w bull y) (x bull z)
This relation between the two operations can be expressed as
w x
y z=
w x
y z=
w x
y z
6 56
The surprising Eckmann-Hilton argument (1)
Assume that and bull satisfy the interchange law
(w x) bull (y z) = (w bull y) (x bull z)
Assume that and bull have identity elements 1 and 1bull respectively
In the interchange law take w = z = 1 and x = y = 1bull
(lowast) (1 1bull) bull (1bull 1) = (1 bull 1bull) (1bull bull 1)
By definition of identity elements
1 1bull = 1bull 1 = 1bull 1 bull 1bull = 1bull bull 1 = 1
Using these to simplify equation (lowast) gives 1bull bull 1bull = 1 1
By definition of identity elements this implies that 1 = 1bull
The two identity elements are equal
7 56
The surprising Eckmann-Hilton argument (2)
Write 1 for the common identity element 1 = 1 = 1bull
In the interchange law take w = z = 1
(1 x) bull (y 1) = (1 bull y) (x bull 1)
By definition of identity element this implies that x bull y = y x
The black operation is the opposite of the white operation
Using this to simplify the interchange law gives
(y z) (w x) = (y w) (z x)
Taking x = y = 1 in the last equation gives
(1 z) (w 1) = (1 w) (z 1)
By definition of identity element this implies that z w = w z
The white operation is commutative
8 56
The surprising Eckmann-Hilton argument (3)
But if x bull y = y x and y x = x y then x bull y = x y
The two operations are equal (There is really only one operation)
Hence the interchange law can be simplified to
(w x) (y z) = (w y) (x z)
Taking x = 1 gives
(w 1) (y z) = (w y) (1 z)
By definition of identity element this implies
w (y z) = (w y) z
The operation is associative
We have proved the (in)famous Eckmann-Hilton Theorem
9 56
The Eckmann-Hilton Theorem
Theorem (Eckmann-Hilton)
Let S be a set with two binary operations S times S rarr S denoted and bullSuppose that these two operations satisfy the interchange lawSuppose also that these two operations have identity elements 1 and 1bullThen the two identity elements are equal the two operations are equaland the single remaining operation is both commutative and associative
Beno Eckmann Peter HiltonGroup-like structures in general categories IMultiplications and comultiplicationsMathematische Annalen 145 (196162) 227ndash255
bull Theorem 333 (page 236)
bull The definition of H-structure (page 241)
bull Theorem 417 (page 244)
10 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
A crossword puzzle is this higher-dimensional algebra
5 56
The interchange law for two operations
Consider the 2times 2 square product y zw x of four variables
There are two different ways to write this using and bull
(w x) bull (y z) or (w bull y) (x bull z)
Note the transpositions of x and y and of the operations and bull
We assume these two ways of writing y zw x always give the same result
This assumption is called the interchange law for all w x y z we have
(w x) bull (y z) = (w bull y) (x bull z)
This relation between the two operations can be expressed as
w x
y z=
w x
y z=
w x
y z
6 56
The surprising Eckmann-Hilton argument (1)
Assume that and bull satisfy the interchange law
(w x) bull (y z) = (w bull y) (x bull z)
Assume that and bull have identity elements 1 and 1bull respectively
In the interchange law take w = z = 1 and x = y = 1bull
(lowast) (1 1bull) bull (1bull 1) = (1 bull 1bull) (1bull bull 1)
By definition of identity elements
1 1bull = 1bull 1 = 1bull 1 bull 1bull = 1bull bull 1 = 1
Using these to simplify equation (lowast) gives 1bull bull 1bull = 1 1
By definition of identity elements this implies that 1 = 1bull
The two identity elements are equal
7 56
The surprising Eckmann-Hilton argument (2)
Write 1 for the common identity element 1 = 1 = 1bull
In the interchange law take w = z = 1
(1 x) bull (y 1) = (1 bull y) (x bull 1)
By definition of identity element this implies that x bull y = y x
The black operation is the opposite of the white operation
Using this to simplify the interchange law gives
(y z) (w x) = (y w) (z x)
Taking x = y = 1 in the last equation gives
(1 z) (w 1) = (1 w) (z 1)
By definition of identity element this implies that z w = w z
The white operation is commutative
8 56
The surprising Eckmann-Hilton argument (3)
But if x bull y = y x and y x = x y then x bull y = x y
The two operations are equal (There is really only one operation)
Hence the interchange law can be simplified to
(w x) (y z) = (w y) (x z)
Taking x = 1 gives
(w 1) (y z) = (w y) (1 z)
By definition of identity element this implies
w (y z) = (w y) z
The operation is associative
We have proved the (in)famous Eckmann-Hilton Theorem
9 56
The Eckmann-Hilton Theorem
Theorem (Eckmann-Hilton)
Let S be a set with two binary operations S times S rarr S denoted and bullSuppose that these two operations satisfy the interchange lawSuppose also that these two operations have identity elements 1 and 1bullThen the two identity elements are equal the two operations are equaland the single remaining operation is both commutative and associative
Beno Eckmann Peter HiltonGroup-like structures in general categories IMultiplications and comultiplicationsMathematische Annalen 145 (196162) 227ndash255
bull Theorem 333 (page 236)
bull The definition of H-structure (page 241)
bull Theorem 417 (page 244)
10 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
The interchange law for two operations
Consider the 2times 2 square product y zw x of four variables
There are two different ways to write this using and bull
(w x) bull (y z) or (w bull y) (x bull z)
Note the transpositions of x and y and of the operations and bull
We assume these two ways of writing y zw x always give the same result
This assumption is called the interchange law for all w x y z we have
(w x) bull (y z) = (w bull y) (x bull z)
This relation between the two operations can be expressed as
w x
y z=
w x
y z=
w x
y z
6 56
The surprising Eckmann-Hilton argument (1)
Assume that and bull satisfy the interchange law
(w x) bull (y z) = (w bull y) (x bull z)
Assume that and bull have identity elements 1 and 1bull respectively
In the interchange law take w = z = 1 and x = y = 1bull
(lowast) (1 1bull) bull (1bull 1) = (1 bull 1bull) (1bull bull 1)
By definition of identity elements
1 1bull = 1bull 1 = 1bull 1 bull 1bull = 1bull bull 1 = 1
Using these to simplify equation (lowast) gives 1bull bull 1bull = 1 1
By definition of identity elements this implies that 1 = 1bull
The two identity elements are equal
7 56
The surprising Eckmann-Hilton argument (2)
Write 1 for the common identity element 1 = 1 = 1bull
In the interchange law take w = z = 1
(1 x) bull (y 1) = (1 bull y) (x bull 1)
By definition of identity element this implies that x bull y = y x
The black operation is the opposite of the white operation
Using this to simplify the interchange law gives
(y z) (w x) = (y w) (z x)
Taking x = y = 1 in the last equation gives
(1 z) (w 1) = (1 w) (z 1)
By definition of identity element this implies that z w = w z
The white operation is commutative
8 56
The surprising Eckmann-Hilton argument (3)
But if x bull y = y x and y x = x y then x bull y = x y
The two operations are equal (There is really only one operation)
Hence the interchange law can be simplified to
(w x) (y z) = (w y) (x z)
Taking x = 1 gives
(w 1) (y z) = (w y) (1 z)
By definition of identity element this implies
w (y z) = (w y) z
The operation is associative
We have proved the (in)famous Eckmann-Hilton Theorem
9 56
The Eckmann-Hilton Theorem
Theorem (Eckmann-Hilton)
Let S be a set with two binary operations S times S rarr S denoted and bullSuppose that these two operations satisfy the interchange lawSuppose also that these two operations have identity elements 1 and 1bullThen the two identity elements are equal the two operations are equaland the single remaining operation is both commutative and associative
Beno Eckmann Peter HiltonGroup-like structures in general categories IMultiplications and comultiplicationsMathematische Annalen 145 (196162) 227ndash255
bull Theorem 333 (page 236)
bull The definition of H-structure (page 241)
bull Theorem 417 (page 244)
10 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
The surprising Eckmann-Hilton argument (1)
Assume that and bull satisfy the interchange law
(w x) bull (y z) = (w bull y) (x bull z)
Assume that and bull have identity elements 1 and 1bull respectively
In the interchange law take w = z = 1 and x = y = 1bull
(lowast) (1 1bull) bull (1bull 1) = (1 bull 1bull) (1bull bull 1)
By definition of identity elements
1 1bull = 1bull 1 = 1bull 1 bull 1bull = 1bull bull 1 = 1
Using these to simplify equation (lowast) gives 1bull bull 1bull = 1 1
By definition of identity elements this implies that 1 = 1bull
The two identity elements are equal
7 56
The surprising Eckmann-Hilton argument (2)
Write 1 for the common identity element 1 = 1 = 1bull
In the interchange law take w = z = 1
(1 x) bull (y 1) = (1 bull y) (x bull 1)
By definition of identity element this implies that x bull y = y x
The black operation is the opposite of the white operation
Using this to simplify the interchange law gives
(y z) (w x) = (y w) (z x)
Taking x = y = 1 in the last equation gives
(1 z) (w 1) = (1 w) (z 1)
By definition of identity element this implies that z w = w z
The white operation is commutative
8 56
The surprising Eckmann-Hilton argument (3)
But if x bull y = y x and y x = x y then x bull y = x y
The two operations are equal (There is really only one operation)
Hence the interchange law can be simplified to
(w x) (y z) = (w y) (x z)
Taking x = 1 gives
(w 1) (y z) = (w y) (1 z)
By definition of identity element this implies
w (y z) = (w y) z
The operation is associative
We have proved the (in)famous Eckmann-Hilton Theorem
9 56
The Eckmann-Hilton Theorem
Theorem (Eckmann-Hilton)
Let S be a set with two binary operations S times S rarr S denoted and bullSuppose that these two operations satisfy the interchange lawSuppose also that these two operations have identity elements 1 and 1bullThen the two identity elements are equal the two operations are equaland the single remaining operation is both commutative and associative
Beno Eckmann Peter HiltonGroup-like structures in general categories IMultiplications and comultiplicationsMathematische Annalen 145 (196162) 227ndash255
bull Theorem 333 (page 236)
bull The definition of H-structure (page 241)
bull Theorem 417 (page 244)
10 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
The surprising Eckmann-Hilton argument (2)
Write 1 for the common identity element 1 = 1 = 1bull
In the interchange law take w = z = 1
(1 x) bull (y 1) = (1 bull y) (x bull 1)
By definition of identity element this implies that x bull y = y x
The black operation is the opposite of the white operation
Using this to simplify the interchange law gives
(y z) (w x) = (y w) (z x)
Taking x = y = 1 in the last equation gives
(1 z) (w 1) = (1 w) (z 1)
By definition of identity element this implies that z w = w z
The white operation is commutative
8 56
The surprising Eckmann-Hilton argument (3)
But if x bull y = y x and y x = x y then x bull y = x y
The two operations are equal (There is really only one operation)
Hence the interchange law can be simplified to
(w x) (y z) = (w y) (x z)
Taking x = 1 gives
(w 1) (y z) = (w y) (1 z)
By definition of identity element this implies
w (y z) = (w y) z
The operation is associative
We have proved the (in)famous Eckmann-Hilton Theorem
9 56
The Eckmann-Hilton Theorem
Theorem (Eckmann-Hilton)
Let S be a set with two binary operations S times S rarr S denoted and bullSuppose that these two operations satisfy the interchange lawSuppose also that these two operations have identity elements 1 and 1bullThen the two identity elements are equal the two operations are equaland the single remaining operation is both commutative and associative
Beno Eckmann Peter HiltonGroup-like structures in general categories IMultiplications and comultiplicationsMathematische Annalen 145 (196162) 227ndash255
bull Theorem 333 (page 236)
bull The definition of H-structure (page 241)
bull Theorem 417 (page 244)
10 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
The surprising Eckmann-Hilton argument (3)
But if x bull y = y x and y x = x y then x bull y = x y
The two operations are equal (There is really only one operation)
Hence the interchange law can be simplified to
(w x) (y z) = (w y) (x z)
Taking x = 1 gives
(w 1) (y z) = (w y) (1 z)
By definition of identity element this implies
w (y z) = (w y) z
The operation is associative
We have proved the (in)famous Eckmann-Hilton Theorem
9 56
The Eckmann-Hilton Theorem
Theorem (Eckmann-Hilton)
Let S be a set with two binary operations S times S rarr S denoted and bullSuppose that these two operations satisfy the interchange lawSuppose also that these two operations have identity elements 1 and 1bullThen the two identity elements are equal the two operations are equaland the single remaining operation is both commutative and associative
Beno Eckmann Peter HiltonGroup-like structures in general categories IMultiplications and comultiplicationsMathematische Annalen 145 (196162) 227ndash255
bull Theorem 333 (page 236)
bull The definition of H-structure (page 241)
bull Theorem 417 (page 244)
10 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
The Eckmann-Hilton Theorem
Theorem (Eckmann-Hilton)
Let S be a set with two binary operations S times S rarr S denoted and bullSuppose that these two operations satisfy the interchange lawSuppose also that these two operations have identity elements 1 and 1bullThen the two identity elements are equal the two operations are equaland the single remaining operation is both commutative and associative
Beno Eckmann Peter HiltonGroup-like structures in general categories IMultiplications and comultiplicationsMathematische Annalen 145 (196162) 227ndash255
bull Theorem 333 (page 236)
bull The definition of H-structure (page 241)
bull Theorem 417 (page 244)
10 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
A brief digression on homotopy groups
The homotopy groups for n ge 1 of a topological space are defined in termsof continuous maps of the n-sphere into the spaceThe first homotopy group (the fundamental group) is defined in terms ofloops in the space and is usually noncommutative
Corollary (of the Eckmann-Hilton Theorem)
For n ge 2 the higher homotopy groups are always commutative
Ronald Brown groupoidsorguk (slightly edited quotation)
ldquoThe nonabelian fundamental group gave more information than the firsthomology group Topologists were seeking higher dimensional versions ofthe fundamental group Cech submitted to the 1932 ICM a paper onhigher homotopy groups These were quickly proved to be abelian indimensions gt 1 and Cech was asked (by Alexandrov and Hopf) towithdraw his paper Only a short paragraph appeared in the Proceedingsrdquo
11 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Familiar examples of the interchange law (1)
Example (multiplication and division)
Suppose that the underlying set is the nonzero real numbers R 0Let the operations and bull be multiplication and division respectively
Then the interchange law states that
(w middot x)(y middot z) = (wy) middot (xz)
which is simply the familiar rule for multiplying fractions
w middot xy middot z
=w
ymiddot xz
(Why does the Eckmann-Hilton Theorem not imply that multiplicationand division are the same commutative associative operation)
12 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Familiar examples of the interchange law (2)
Example (composition of functions on Cartesian products)
Suppose that denotes the operation of forming an ordered pair
f1 f2 = (f1 f2)
Suppose that bull denotes the operation of function composition
(f bull g)(minus) = f (g(minus))
What does the interchange law state in this setting
(f1 f2) bull (g1 g2) = (f1 bull g1) (f2 bull g2) lArrrArr((f1 f2) bull (g1 g2)
)(minusminus)
)=(
(f1 bull g1)(minus) f2 bull g2)(minus))
The interchange law shows how to define composition of Cartesianproducts of functions on Cartesian products of sets
13 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Familiar examples of the interchange law (3)
Example (endomorphism PROP of a set)
Let X be a nonempty set and let X n be its n-th Cartesian power (n ge 1)
Let Map(m n) be the set of all functions f Xm rarr X n for m n ge 1
Let Map(X ) be the disjoint union of the sets Map(m n)
Map(X ) =⊔
mnge1
Map(m n)
On Map(X ) there are two natural binary operations
The horizontal product For f X p rarr X q and g X r rarr X s we definethe operation Map(p q)timesMap(r s) minusrarrMap(p+r q+s) as follows
f g X p+r = X p times X r (f g)minusminusminusminusminusminusrarr X q times X s = X q+s
This operation is defined for all p q r s ge 1
14 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Example (continued)
The vertical product bull For f X q rarr X r and g X p rarr X q we definethe operation bull Map(q r)timesMap(p q) minusrarrMap(p r) as follows
f bull g X p f bull gminusminusminusminusminusminusrarr X r
This operation is defined on Map(q1 r)timesMap(p q2) only if q1 = q2
These two operations satisfy the interchange law
(f g) bull (h k) = (f bull h) (g bull k)
To verify this compare the following maps check the results are equal
(f g) bull (h k) X p times X q (hk)minusminusminusminusminusminusrarr X r times X s (f g)minusminusminusminusminusminusrarr X t times X u
(f bull h) (g bull k) X p times X q ( f bullh gbullk )minusminusminusminusminusminusminusminusminusminusrarr X t times X u
15 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Familiar examples of the interchange law (4)
Definition
A double category D is a pair of categories (D0D1) with functorse D0 rarr D1 (identity map) s t D1 rarr D0 (source and target objects)
In D0 we denote objects by capital Latin letters A (the 0-cells of D)and morphisms by arrows labelled by lower-case italic letters u (the vertical 1-cells of D)
In D1 the objects are arrows labelled by lower-case italic letters h (the horizontal 1-cells of D) and the morphisms are arrows labelled bylower-case Greek letters α β (the 2-cells of D)
If A is an object in D0 then e(A) is the (horizontal) identity arrow on A(By Eckmann-Hilton identity arrows may exist in only one direction)
The functors s t are source and target if h is a horizontal arrow in D1
then s(h) and t(h) are its domain and codomain objects in D0
These functors satisfy the equation s(e(A)) = t(e(A)) = A for every A
16 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Definition
If α β are 2-cells then horizontal and vertical composition α β α βare defined by the following diagrams (and satisfy the interchange law)
A C E
B D F
u
OOv
OOw
OO
h
k
` m
α
KS
β
KS
sim=A E
B F
u
OOw
OO
kh
m`
α β
KS
A
B
C
D
E
F
u
OO
v
OO
w
OO
x
OO
h
`
α
KS
β
KS
sim=A
C
D
F
h
`
vuOO
xwOO
αβ
KS
17 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Introductory references on higher-dimensional algebra
bull J C Baez An introduction to n-categories 7th Conf on CategoryTheory and Computer Science (1997) pages 1ndash33 Lecture Notes inComputer Science 1290
bull J C Baez M Stay Physics topology logic and computation aRosetta Stone New Structures for Physics (2011) pages 95ndash172 LectureNotes in Physics 813
bull M A Batanin The Eckman-Hilton argument and higher operadsAdv Math 217 (2008) 334ndash385
bull R Brown Out of line Royal Institution Proc 64 (1992) 207ndash243
bull R Brown T Porter Intuitions of higher dimensional algebra forthe study of structured space Revue de Synthese 124 (2003) 173ndash203
bull R Brown T Porter Category theory higher-dimensional algebrapotential descriptive tools in neuroscience Proc International Conferenceon Theoretical Neurobiology Delhi 2003
18 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Nonassociative algebra inone and two dimensions
19 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Nonassociativity for one operation
Associativity can be applied in various ways to a given association typeIn each arity this gives a partially ordered set the Tamari latticeOn the left we have the Tamari lattice of binary trees for n = 4
(((ab)c)d)e((a(bc))d)e((ab)(cd))e(a((bc)d))e(a(b(cd)))e((ab)c)(de)(a(bc))(de)(ab)((cd)e)(ab)(c(de))a(((bc)d)e)a((b(cd))e)a((bc)(de))a(b((cd)e))a(b(c(de)))
On the right association types are represented by dyadic partitions ofthe unit interval [0 1] where multiplication corresponds to bisection
20 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Two nonassociative operations with interchange law (1)
How do we generalize association types to two dimensions
Thinking in terms of placements of parentheses we could use two differenttypes of brackets (minusminus) to represent minus minus and [minusminus] to represent minus bull minus
Thinking in terms of binary trees we could label each internal node(including the root) by one or the other of the operation symbols bullAlternatively in one dimension wersquove seen that association typescorrespond bijectively to dyadic partitions of the unit interval [0 1]
This suggests that in two dimensions we should be able to interpretassociation types as dyadic partitions of the unit square [0 1]times [0 1]
In this case represents the horizontal operation (vertical bisection)and bull represents the vertical operation (horizontal bisection)
This geometric interpretation has the advantage that the two operations(both nonassociative) automatically satisfy the interchange law
21 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Two nonassociative operations with interchange law (2)
Definition
A 2-dimensional association type of arity n is a dyadic partition of the unitsquare [0 1]times [0 1] that is obtained by nminus1 vertical or horizontalbisections of subrectangles starting from the (empty) unit square
n = 1 lowast (lowast is the argument symbol minus takes up too much space)
n = 2 lowastlowast lowastbulllowast
n = 3 (lowastlowast)lowast lowast(lowastlowast) (lowastbulllowast)bulllowast lowastbull(lowastbulllowast)
(lowastbulllowast)lowast lowast(lowastbulllowast) (lowastlowast)bulllowast lowastbull(lowastlowast)
22 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Two-dimensional association types in arity n = 4 (part 1)
23 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Two-dimensional association types in arity n = 4 (part 2)
total 39
24 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Enumeration of two-dimensional association types (1)
The sequence 1 2 8 39 satisfies the following recurrence relationwhich generalizes the familiar recurrence relation for the Catalan numbers
C (1) = 1 C (n) = 2sum
i j C (i)C (j)minussum
i j kl C (i)C (j)C (k)C (l)
First sum is over all 2-compositions of n (i+j = n) into positive integerssecond sum is over all 4-compositions of n (i+j+k+l = n)The generating function for this sequence
G (x) =sum
nge1 C (n)xn
satisfies a quartic polynomial equation which generalizes the familiarquadratic polynomial equation for the Catalan numbers
G (x)4 minus 2G (x)2 + G (x)minus x = 0
Wersquove generalized this to all dimensions (proof by homological algebra)Murray Bremner Vladimir DotsenkoBoardman-Vogt tensor products of absolutely free operadsProceedings A of the Royal Society of Edinburgh (to appear)
25 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Enumeration of two-dimensional association types (2)
The sequence counting 2-dimensional association types starts like this
1 2 8 39 212 1232 7492 47082 303336 1992826
13299624 89912992 614474252 4238138216 29463047072
206234876287 1452319244772 (oeisorgA236339)
The number of dyadic partitions of the unit square into n rectangles is
C22(n) =1
n
b nminus13csum
i=0
(2(nminus1minusi)
nminus1 nminus1minus3i i
)(minus1)i 2nminus1minus3i
Yu Hin (Gary) Au Fatemeh Bagherzadeh Murray Bremner
Enumeration and Asymptotic Formulas forRectangular Partitions of the Hypercube
arXiv190300813v1[mathCO]
26 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
A related application VLSI design floorplanning
alumnisoeucscedu~sloganresearchhtml
27 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
A related application rectangular cartograms
28 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
A related application ldquosquaring the squarerdquo
wwwsquaringnethistory_theoryduijvestijnhtml
29 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
A related application ldquoguillotine partitionsrdquo in 3D
A Asinowski G Barequet T Mansour R PinterCut equivalence of d-dimensional guillotine partitionsDiscrete Mathematics Volume 331 28 September 2014 Pages 165ndash174
30 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
A related application the ldquolittle n-cubesrdquo operad
wwwmath3macommathema20171030what-is-an-operad-part-2
31 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Double interchange semigroupstwo associative operations
satisfying interchange(a toy model of double categories)
32 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
What about two associative operations related by the interchange law
An unexpected commutativity relation in arity 16 was discovered by
Joachim KockCommutativity in double semigroups and two-fold monoidal categoriesJ Homotopy and Related Structures 2 (2007) no 2 217ndash228
A former postdoctoral fellow and I used computer algebra (Maple) to showthat arity 9 is the lowest in which such commutativity relations occur
Murray Bremner Sara MadariagaPermutation of elements in double semigroupsSemigroup Forum 92 (2016) no 2 335ndash360
Further relations in arity 10 were discovered by my current postdoc
Fatemeh Bagherzadeh Murray BremnerCommutativity in double interchange semigroupsApplied Categorical Structures 26 (2018) no 6 1185ndash1210
33 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Kockrsquos surprising observation
Relation of arity 16 associativity and the interchange law combine toimply a commutativity relation the equality of two monomials withminus same skeleton (placement of parentheses and operation symbols)minus different permutations of arguments (transposition of f g)
(a2 b2 c 2 d) (e 2 f 2 g 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q) equiv(a2 b2 c 2 d) (e 2 g 2 f 2 h) (i 2 j 2 k 2 `) (m2 n2 p 2 q)
a b c d
e f g h
i j k `
m n p q
equiv
a b c d
e g f h
i j k `
m n p q
The symbol equiv indicates that the equation holds for all arguments
34 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Nine is the least arity for a commutativity relation
((a2 b)2 c) (((d 2 (e f ))2 (g h))2 i) equiv((a2 b)2 c) (((d 2 (g f ))2 (e h))2 i)
a b c
de
f
g
hi
equiv
a b c
de
f
g
hi
35 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
a b f
d
h
cg
e ij
a b f
d
h
cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Geometric proof of commutativity in arity 10
Watch what happens to the arguments d and g as we repeatedly applyassociativity horizontal and vertically together with the interchange law
At the end we have exactly the same dyadic partition of the square butthe arguments d and g have changed places
a bf
gh
cd
ei j
a b f
g
h
cd
e ij
a b f
g
h
cd
e ij
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
ei
j
ab
f
g
h
c
d
e i j
36 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
h
c
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ei j
ab
f
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ei j
a b f
g
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ab
c
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f
g
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a b f
g
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cd
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cd
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a b f
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cg
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cg
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a bf
dh
cg
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37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
ab
f
g
h
c
d
e i j
ab
f
g
h
c
d
e i j
ab
f
g
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d
e i j
ab
f
g
h
c
d
ei j
ab
f
g
h
c
d
ei j
a b f
g
h
c
d
ei j
ab
c
d
e
f
g
h
i j
a b f
g
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cd
e ij
a b f
g
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cd
e ij
a b f
d
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cg
e ij
a b f
d
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cg
e ij
a bf
dh
cg
ei j
37 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Four nonassociative operads Free Inter BP DBP
Definition
Free free symmetric operad two binary operations with no symmetrydenoted M (horizontal) and N (vertical)
Definition
Inter quotient of Free by ideal I = 〈〉 generated by interchange law
(a M b) N (c M d)minus (a N c) M (b N d) equiv 0
Definition
BP set operad of block partitions of open unit square I 2 I = (0 1)Horizontal composition x rarr y (vertical composition x uarr y)
bull translate y one unit east (north) to get y + ei (i = 1 2)
bull form x cup (y + ei ) to get a partition of width (height) two
bull scale horizontally (vertically) by one-half to get a partition of I 2
38 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Algorithm
In dimension d to get a dyadic block partition of I d (unit d-cube)
Set P1 larr I d Do these steps for i = 1 kminus1 (k parts)
Choose an empty block B isin Pi and an axis j isin 1 dIf (aj bj) is projection of B onto axis j then set c larr 1
2 (aj+bj)
Set B primeB primeprime larr B x isin B | xj = c (hyperplane bisection)
Set Pi+1 larr (Pi B ) t B primeB primeprime (replace B by B prime B primeprime)
Definition
bull DBP unital suboperad of BP generated by and
bull Unital include unary operation I 2 (block partition with one empty block)
bull DBP consists of dyadic block partitions
every P isin DBP with n+1 parts is obtained from some Q isin DBPwith n parts by bisection of a part of Q horizontally or vertically
39 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Three associative operads AssocB AssocNB DIA
Definition
AssocB quotient of Free by ideal A = 〈AM AN 〉 generated by
AM (a b c) = ( a M b ) M c minus a M ( b M c ) (horizontal associativity)
AN (a b c) = ( a N b ) N c minus a N ( b N c ) (vertical associativity)
AssocNB isomorphic copy of AssocB with following change of basis
ρ AssocBrarr AssocNB represents rewriting a coset representative(binary tree) as a nonbinary (= not necessarily binary) tree
new basis consists of disjoint union x1 t TM t TNisolated leaf x1 and two copies of TT = all labelled rooted plane trees with at least one internal node
TM root r of every tree has label M labels alternate by level
TN labels of internal nodes (including root) are reversed
40 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Algorithm for converting binary tree to nonbinary tree
We write Assoc if convenient for AssocB sim= AssocNB
M
M M
T1T2T3T4
αminusminusminusrarr
M
T1T2T3T4
M
M N
T1T2T3T4
αminusminusminusrarr
M
NT1T2
T3T4
M
N M
T1T2T3T4
αminusminusminusrarr
M
N
T1T2
T3T4
M
N N
T1T2T3T4
αminusminusminusminusminusminusrarrno change
M
N N
T1T2T3T4
Switching M N throughout defines α for subtrees with roots labelled N
41 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Associativity =rArr interchange applies almost everywhere
After converting a binary tree to a nonbinary (not necessarily binary) treeif the root is white (horizontal) then all of its children are black (vertical)all of its grandchildren are white all of its great-grandchildren are blacketc alternating white and black according to the level
M
N middot middot middot N middot middot middot N
M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M M middot middot middot M middot middot middot M
N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN N middot middot middotN
M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotM M middot middot middotMIf the root is black then we simply transpose white and black throughout
42 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Definition
DIA quotient of Free by ideal 〈AM AN 〉This is the algebraic operad governing double interchange algebraswhich possess two associative operations satisfying the interchange law
bull Inter AssocB AssocNB DIA are defined by relations v1 minus v2 equiv 0(equivalently v1 equiv v2) where v1 v2 are cosets of monomials in Freebull We could work with set operads (we never need linear combinations)bull Vector spaces and sets are connected by a pair of adjoint functors
the forgetful functor sending a vector space V to its underlying setthe left adjoint sending a set S to the vector space with basis S bull Corresponding relation between Grobner bases and rewrite systems
if we compute a syzygy for two tree polynomials v1 minus v2 and w1 minus w2then the common multiple of the leading terms cancelsand we obtain another difference of tree monomialssimilarly from a critical pair of rewrite rules v1 7rarr v2 and w1 7rarr w2we obtain another rewrite rule
43 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Morphisms between operads
bull Our goal is to understand the operad DIAbull We have no convenient normal form for the basis monomials of DIAbull There is a normal form if we factor out associativity but not interchangebull There is a normal form if we factor out interchange but not associativitybull We use the monomial basis of the operad Freebull We apply rewrite rules which express associativity of each operation
(right to left or reverse) and interchange between the operations(black to white or reverse)bull These rewritings convert one monomial in Free to another monomial
which is equivalent to the first modulo associativity and interchangebull Given an element X of DIA represented by a monomial T in Free
we convert T to another monomial T prime in the same inverse image as Twith respect to the natural surjection Free DIAbull We use undirected rewriting to pass from T to T prime we may need to
reassociate left to right apply interchange reassociate right to left
44 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Commutative diagram of operads and morphisms
Free
BP
DBP
Inter
AssocB
AssocNB
DIA
minus〈AM AN 〉α
33 33
minus〈〉χ
++ ++
Γ
γ isomorphism
minus〈AM+IAN+I〉
α
33 33
ρ isomorphism
OO
minus〈+A〉χ ++ ++
inclusionι
45 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Further details(if I havenrsquot run out of time already)
46 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Cuts and slices
Definition
bull Subrectangle any union of empty blocks forming a rectanglebull Let P be a block partition of I 2 and let R be a subrectangle of Pbull A main cut in R is a horizontal or vertical bisection of Rbull Every subrectangle has at most two main cuts (horizontal vertical)bull Suppose that a main cut partitions R into subrectangles R1 and R2bull If either R1 or R2 has a main cut parallel to the main cut of R we call
this a primary cut in R we also call the main cut of R a primary cutbull In general if a subrectangle S of R is obtained by a sequence of cuts
parallel to a main cut of R then a main cut of S is a primary cut of Rbull Let C1 C` be the primary cuts of R parallel to a given main cut Ci
(1 le i le `) in positive order (left to right or bottom to top) so thatthere is no primary cut between Cj and Cj+1 for 1 le j le `minus1bull Define ldquocutsrdquo C0 C`+1 to be left right (bottom top) sides of Rbull Write Sj for the j-th slice of R parallel to the given main cut
47 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Commutativity relations
Definition
Suppose that for some monomial m of arity n in the operad Free andfor some transposition (ij) isin Sn the corresponding cosets in DIA satisfy
m(x1 xi xj xn) equiv m(x1 xj xi xn)
In this case we say that m admits a commutativity relation
Proposition
(Fatemeh Bagherzadeh) Assume that m is a monomial in Free admittinga commutativity relation which is not a consequence of a commutativityrelation holding in (i) a proper factor of m or (ii) a proper quotient of m(Quotient refers to substitution of a decomposable factor for the sameindecomposable argument on both sides of a relation of lower arity)Then the dyadic block partition P = Γ(m) contains both main cutsIn other words it must be possible to apply the interchange law as arewrite rule at the root of the monomial m (regarded as a binary tree)
48 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Border blocks and interior blocks
Definition
Let P be a block partition of I 2 consisting of empty blocks R1 Rk If the closure of Ri has nonempty intersection with the four sides of theclosure I 2 then Ri is a border block otherwise Ri is an interior block
Lemma
Suppose that P1 = Γ(m1) and P2 = Γ(m2) are two labelled dyadic blockpartitions of I 2 such that m1 equiv m2 in every double interchange semigroupThen any interior (border) block of P1 is an interior (border) block of P2
Lemma
If m admits a commutativity relation then in the corresponding blockpartition P = Γ(m) the two commuting empty blocks are interior blocks
Basic idea of the proofs neither associativity nor the interchange law canchange an interior block to a border block or conversely
49 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Lower bounds on the arity of a commutativity relation
Lemma
If m admits a commutativity relation then P = Γ(m) has both main cutshence P is the union of 4 subsquares A1 A4 (NW NE SW SE)If a subsquare has 1 (2) empty interior block(s) then that subsquare hasat least 3 (4) empty blocks Hence P contains at least 7 empty blocks
Proposition
(Fatemeh Bagherzadeh) If the monomial m of arity n in the operad Freeadmits a commutativity relation then P = Γ(m) has n ge 8 empty blocks
Reflecting P in the horizontal andor vertical axes if necessary we mayassume that the NW subsquare A1 has two empty interior blocks and hasonly the horizontal main cut (otherwise we reflect in the NW-SE diagonal)
50 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
We display the 3 partitions with 7 empty blocks satisfying these conditions
a b
ec d
fg
a b
c d e
fg
a b
c d e
fg
None of these configurations admits a commutativity relation
The method used for the proof of the last proposition can be extended toshow that a monomial of arity 8 cannot admit a commutativity relationalthough the proof is rather long owing to the large number of cases(a) 1 square Ai has 5 empty blocks and the other 3 squares are empty(b) 1 square Ai has 4 empty blocks another square Aj has 2 and theother 2 squares are empty (2 subcases Ai Aj share edge or only corner)(c) 2 squares Ai Aj each have 3 empty blocks other 2 empty (2 subcases)This provides a different proof independent of machine computation ofthe minimality result of Bremner and Madariaga
51 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Commutative block partitions in arity 10
Lemma
Let m admit a commutativity relation in arity 10 Then P = Γ(m) has atleast two and at most four parallel slices in either direction
Proof
By the lemmas P contains both main cuts Since P contains 10 emptyblocks it has at most 5 parallel slices (4 primary cuts) in either directionIf there are 4 primary cuts in one direction and the main cut in the otherdirection then there are 10 empty blocks and all are border blocks
In what follows m has arity 10 and admits a commutativity relationHence P = Γ(m) is a dyadic block partition with 10 empty blocksCommuting blocks are interior P has either 2 3 or 4 parallel slicesIf P has 3 (resp 4) parallel slices then commuting blocks are in middleslice (resp middle 2 slices) Interchanging H and V if necessary we mayassume parallel slices are vertical
52 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Four parallel vertical slices
We have H and V main cuts and 2 more vertical primary cuts
Applying horizontal associativity gives 2 rows of 4 equal empty blocks
This configuration has 8 empty blocks all of which are border blocks
We need 2 more cuts to create 2 interior blocks
Applying vertical associativity in the second slice from the left andapplying a dihedral symmetry of the square (if necessary) reducesthe number of configurations to the following A B C
A B C
53 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
54 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Theorem
Configuration A In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM b)N (c M (d N e)))M (((f N g)M h)N (i M j)) equiv((aM b)N (c M (g N e)))M (((f N d)M h)N (i M j))
For configuration B we label only the two blocks which transposeApplications of associativity and interchange can easily be recovered
cg
cg cg cg c
g
g
c
gc
gc
g
c
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Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
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Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Theorem
Configuration B In every double interchange semigroup the followingcommutativity relation holds for all values of the arguments a j
((aM (bN c))N (f M (g N h)))M ((d M e)N (i M j)) equiv((aM (bN g))N (f M (c N h)))M ((d M e)N (i M j))
For configuration C we obtain no new commutativity relations
Concluding remarks higher dimensions
We have studied structures with two operations representing orthogonal(horizontal and vertical) compositions in two dimensions
Most of our constructions make sense for any number of dimensions d ge 2
Major obstacle for d ge 3 monomial basis for AssocNB consisting ofnonbinary trees with alternating white and black internal nodes does notgeneralize in a straightforward way
55 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
Thanks to all of you for your attentionMerci a vous tous pour votre attention
Gracias a todos por su atencionGracies a tots per la vostra atencio
Eskerrik asko guztioi zuen arretarengatikObrigado a todos por sua atencao
Ngibonga nonke ngokunaka kwakho
56 56
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