Polyadic Systems and their Multiplace Representations STEVEN DUPLIJ Institute of Mathematics, University of M ¨ unster http://wwwmath.uni-muenster.de/u/duplij 2014 1
Jun 22, 2015
Polyadic Systems and their
Multiplace Representations
STEVEN DUPLIJ
Institute of Mathematics, University of Munster
http://wwwmath.uni-muenster.de/u/duplij
2014
1
1 History
1 History
Ternary algebraic operations (with the arity n = 3) were introduced
by A. Cayley in 1845 and later by J. J. Sylvester in 1883.
The notion of an n-ary group was introduced in 1928 by DORNTE
[1929] (inspired by E. Nother).
The coset theorem of Post explained the connection between n-ary
groups and their covering binary groups POST [1940].
The next step in study of n-ary groups was the Gluskin-Hosszu
theorem HOSSZU [1963], GLUSKIN [1965].
The cubic and n-ary generalizations of matrices and determinants
were made in KAPRANOV ET AL. [1994], SOKOLOV [1972], physical
application in KAWAMURA [2003], RAUSCH DE TRAUBENBERG [2008].
2
1 History
Particular questions of ternary group representations wereconsidered, by the SD in BOROWIEC ET AL. [2006], DUPLIJ [2013a].
Some theorems connecting representations of binary and n-arygroups were presented in DUDEK AND SHAHRYARI [2012].
In physics, the most applicable structures are the nonassociativeGrassmann, Clifford and Lie algebras LOHMUS ET AL. [1994],GEORGI [1999]. The ternary analog of Clifford algebra wasconsidered in ABRAMOV [1995], and the ternary analog ofGrassmann algebra ABRAMOV [1996] was exploited to constructternary extensions of supersymmetry ABRAMOV ET AL. [1997].
In the higher arity studies, the standard Lie bracket is replaced by alinear n-ary bracket, and the algebraic structure of thecorresponding model is defined by the additional characteristicidentity for this generalized bracket, corresponding to the Jacobiidentity DE AZCARRAGA AND IZQUIERDO [2010].
3
1 History
The infinite-dimensional version of n-Lie algebras are the Nambu
algebras NAMBU [1973], TAKHTAJAN [1994], and their n-bracket is
given by the Jacobian determinant of n functions, the Nambu
bracket, which in fact satisfies the Filippov identity FILIPPOV [1985].
Ternary Filippov algebras were successfully applied to a
three-dimensional superconformal gauge theory describing the
effective worldvolume theory of coincident M2-branes of M -theory
BAGGER AND LAMBERT [2008a,b], GUSTAVSSON [2009].
4
2 Plan
2 Plan
1. Classification of general polyadic systems and special elements.
2. Definition of n-ary semigroups and groups.
3. Homomorphisms of polyadic systems.
4. The Hosszu-Gluskin theorem and its “q-deformed”
generalization.
5. Multiplace generalization of homorphisms - heteromorpisms.
6. Associativity quivers.
7. Multiplace representations and multiactions.
8. Examples of matrix multiplace representations for ternary
groups.
5
3 Notations
3 Notations
Let G be a underlying set, universe, carrier, gi ∈ G.
The n-tuple (or polyad) g1, . . . , gn is denoted by (g1, . . . , gn).
The Cartesian product G×n consists of all n-tuples (g1, . . . , gn).
For equal elements g ∈ G, we denote n-tuple (polyad) by (gn).
If the number of elements in the n-tuple is clear from the context or
is not important, we denote it with one bold letter (g), or(g(n)
).
The i-projection Pr(n)i : G×n → G is (g1, . . . gi, . . . , gn) 7−→ gi.
The i-diagonal Diagn : G→ G×n sends one element to the equal
element n-tuple g 7−→ (gn).
6
3 Notations
The one-point set {•} is a “unit” for the Cartesian product, since
there are bijections between G and G× {•}×n, and denote it by ε.
On the Cartesian product G×n one can define a polyadic (n-ary,
n-adic, if it is necessary to specify n, its arity or rank) operation
μn : G×n → G.
For operations we use Greek letters and square brackets μn [g].
The operations with n = 1, 2, 3 are called unary, binary and ternary.
The case n = 0 is special and corresponds to fixing a distinguished
element of G, a “constant” c ∈ G, it is called a 0-ary operation μ(c)0 ,
which maps the one-point set {•} to G, such that μ(c)0 : {•} → G,
and formally has the value μ(c)0 [{•}] = c ∈ G. The 0-ary operation
“kills” arity BERGMAN [1995]
μn+m−1 [g,h] = μn [g, μm [h]] . (3.1)
7
3 Notations
Then, if to compose μn with the 0-ary operation μ(c)0 , we obtain
μ(c)n−1 [g] = μn [g, c] , (3.2)
because g is a polyad of length (n− 1). Visually, it is seen from the
commutative diagram
G×(n−1) × {•}id×(n−1) ×μ(c)0 G×n
G×(n−1)
ε
μ(c)n−1
G
μn (3.3)
which is a definition of a new (n− 1)-ary operation μ(c)n−1.
Remark 3.1. It is important to make a clear distinction between the
0-ary operation μ(c)0 and its value c in G.
8
4 Preliminaries
4 Preliminaries
Definition 4.1. A polyadic system G is a set G together with
polyadic operations, which is closed under them.
Here, we mostly consider concrete polyadic systems with one
“chief” (fundamental) n-ary operation μn, which is called polyadic
multiplication (or n-ary multiplication).
Definition 4.2. A n-ary system Gn = 〈G | μn〉 is a set G closed
under one n-ary operation μn (without any other additional
structure).
Let us consider the changing arity problem:
Definition 4.3. For a given n-ary system 〈G | μn〉 to construct
another polyadic system 〈G | μ′n′〉 over the same set G, which has
multiplication with a different arity n′.
9
4 Preliminaries
There are 3 ways to change arity of operation:
1. Iterating. Using composition of the operation μn with itself, one
can increase the arity from n to n′iter . Denote the number of
iterating multiplications by `μ, and use the bold Greek letters
μ`μn for the resulting composition of n-ary multiplications
μ′n′ = μ`μn
def=
`μ︷ ︸︸ ︷
μn ◦(μn ◦ . . .
(μn × id
×(n−1)). . .× id×(n−1)
),
(4.1)
where n′ = niter = `μ (n− 1) + 1, which gives the length of a
polyad (g) in the notation μ`μn [g]. The operation μ`μn is named
a long product DORNTE [1929] or derived DUDEK [2007].
10
4 Preliminaries
2. Reducing (Collapsing). Using nc distinguished elements orconstants (or nc additional 0-ary operations μ(ci)0 , i = 1, . . . nc),one can decrease arity from n to n′red (as in (3.2)), such thata
μ′n′ = μ(c1...cnc )n′
def= μn ◦
nc︷ ︸︸ ︷μ(c1)0 × . . .× μ(cnc )0 × id×(n−nc)
,
(4.2)where
n′ = nred = n− nc, (4.3)
and the 0-ary operations μ(ci)0 can be on any places.
3. Mixing. Changing (increasing or decreasing) arity may be doneby combining iterating and reducing (maybe with additionaloperations of different arity).
aIn DUDEK AND MICHALSKI [1984] μ(c1...cnc )n is named a retract (which term is
already busy and widely used in category theory for another construction).
11
5 Special elements and properties of n-ary systems
5 Special elements and properties of n-ary systems
Definition 5.1. A zero
μn [g, z] = z, (5.1)
where z can be on any place in the l.h.s. of (5.1).
Only one zero (if its place is not fixed) can be possible in a polyadic
system.
An analog of positive powers of an element POST [1940] should
coincide with the number of multiplications `μ in the iterating (4.1).
Definition 5.2. A (positive) polyadic power of an element is
g〈`μ〉 = μ`μn
[g`μ(n−1)+1
]. (5.2)
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5 Special elements and properties of n-ary systems
Example 5.3. Consider a polyadic version of the binary q-additionwhich appears in study of nonextensive statistics (see, e.g.,TSALLIS [1994], NIVANEN ET AL. [2003])
μn [g] =
n∑
i=1
gi + ~n∏
i=1
gi, (5.3)
where gi ∈ C and ~ = 1− q0, q0 is a real constant ( q0 6= 1 or ~ 6= 0).It is obvious that g〈0〉 = g, and
g〈1〉 = μn
[gn−1, g〈0〉
]= ng + ~gn, (5.4)
g〈k〉 = μn
[gn−1, g〈k−1〉
]= (n− 1) g +
(1 + ~gn−1
)g〈k−1〉. (5.5)
Solving this recurrence formula for we get
g〈k〉 = g
(
1 +n− 1~
g1−n)(1 + ~gn−1
)k−n− 1~
g2−n. (5.6)
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5 Special elements and properties of n-ary systems
Definition 5.4. An element of a polyadic system g is called
`μ-nilpotent (or simply nilpotent for `μ = 1), if there exist such `μthat
g〈`μ〉 = z. (5.7)
Definition 5.5. A polyadic system with zero z is called `μ-nilpotent,
if there exists `μ such that for any (`μ (n− 1) + 1)-tuple (polyad) g
we have
μ`μn [g] = z. (5.8)
Therefore, the index of nilpotency (number of elements whose
product is zero) of an `μ-nilpotent n-ary system is (`μ (n− 1) + 1),
while its polyadic power is `μ .
14
5 Special elements and properties of n-ary systems
Definition 5.6. A polyadic (n-ary) identity (or neutral element) of apolyadic system is a distinguished element ε (and thecorresponding 0-ary operation μ(ε)0 ) such that for any elementg ∈ G we have ROBINSON [1958]
μn[g, εn−1
]= g, (5.9)
where g can be on any place in the l.h.s. of (5.9).
In binary groups the identity is the only neutral element, while inpolyadic systems, there exist many neutral polyads n consisting ofelements of G satisfying
μn [g,n] = g, (5.10)
where g can be also on any place. The neutral polyads are notdetermined uniquely.
The sequence of polyadic identities εn−1 is a neutral polyad.
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5 Special elements and properties of n-ary systems
Definition 5.7. An element of a polyadic system g is called`μ-idempotent (or simply idempotent for `μ = 1), if there exist such`μ that
g〈`μ〉 = g. (5.11)
Both zero and the identity are `μ-idempotents with arbitrary `μ.
We define (total) associativity as invariance of the composition oftwo n-ary multiplications
μ2n [g,h,u] = μn [g, μn [h] ,u] = inv. (5.12)
Informally, “internal brackets/multiplication can be moved on anyplace”, which gives n relations
μn ◦(μn × id
×(n−1))= . . . = μn ◦
(id×(n−1)×μn
). (5.13)
There are many other particular kinds of associativity THURSTON
[1949] and studied in BELOUSOV [1972], SOKHATSKY [1997].
16
5 Special elements and properties of n-ary systems
Definition 5.8. A polyadic semigroup (n-ary semigroup) is a n-ary
system in which the operation is associative, or
Gsemigrpn = 〈G | μn | associativity〉.
In a polyadic system with zero (5.1) one can have trivial
associativity, when all n terms are (5.12) are equal to zero, i.e.
μ2n [g] = z (5.14)
for any (2n− 1)-tuple g.
Proposition 5.9. Any 2-nilpotent n-ary system (having index of
nilpotency (2n− 1)) is a polyadic semigroup.
17
5 Special elements and properties of n-ary systems
It is very important to find the associativity preserving conditions,where an associative initial operation μn leads to an associativefinal operation μ′n′ during the change of arity.
Example 5.10. An associativity preserving reduction can be givenby the construction of a binary associative operation using(n− 2)-tuple c consisting of nc = n− 2 different constants
μ(c)2 [g, h] = μn [g, c, h] . (5.15)
Associativity preserving mixing constructions with different aritiesand places were considered in DUDEK AND MICHALSKI [1984],MICHALSKI [1981], SOKHATSKY [1997].
Definition 5.11. An associative polyadic system with identity (5.9)is called a polyadic monoid.
The structure of any polyadic monoid is fixed POP AND POP [2004]:iterating a binary operation CUPONA AND TRPENOVSKI [1961].
18
5 Special elements and properties of n-ary systems
Several analogs of binary commutativity of polyadic system.
A polyadic system is σ-commutative, if μn = μn ◦ σ
μn [g] = μn [σ ◦ g] , (5.16)
where σ ◦ g =(gσ(1), . . . , gσ(n)
)is a permutated polyad and σ is a
fixed element of Sn. If (5.16) holds for all σ ∈ Sn, then a polyadicsystem is commutative. A special type of the σ-commutativity
μn [g, t, h] = μn [h, t, g] , (5.17)
where t is any fixed (n− 2)-polyad, is called semicommutativity. Sofor a n-ary semicommutative system we have
μn[g, hn−1
]= μn
[hn−1, g
]. (5.18)
If a n-ary semigroup Gsemigrp is iterated from a commutative binarysemigroup with identity, then Gsemigrp is semicommutative.
19
5 Special elements and properties of n-ary systems
Another generalization of commutativity - to generalize binary
mediality in semigroups
(g11 ∙ g12) ∙ (g21 ∙ g22) = (g11 ∙ g21) ∙ (g12 ∙ g22) , (5.19)
following from binary commutativity. For n-ary case they’re different.
Definition 5.12. A polyadic system is medial (or entropic), if
EVANS [1963], BELOUSOV [1972]
μn
μn [g11, . . . , g1n]...
μn [gn1, . . . , gnn]
= μn
μn [g11, . . . , gn1]...
μn [g1n, . . . , gnn]
. (5.20)
The semicommutative polyadic semigroups are medial, as in the
binary case, but, in general (except n = 3) not vice versa GŁAZEK
AND GLEICHGEWICHT [1982].
20
5 Special elements and properties of n-ary systems
Definition 5.13. A polyadic system is cancellative, if
μn [g, t] = μn [h, t] =⇒ g = h, (5.21)
where g, h can be on any place. This means that the mapping μn isone-to-one in each variable. If g, h are on the same i-th place onboth sides, the polyadic system is called i-cancellative.
Definition 5.14. A polyadic system is called (uniquely) i-solvable,if for all polyads t, u and element h, one can (uniquely) resolve theequation (with respect to h) for the fundamental operation
μn [u, h, t] = g (5.22)
where h can be on any i-th place.
Definition 5.15. A polyadic system which is uniquely i-solvable forall places i is called a n-ary (or polyadic)quasigroup.
Definition 5.16. An associative polyadic quasigroup is called an-ary (or polyadic)group.
21
5 Special elements and properties of n-ary systems
In a polyadic group the only solution of (5.22) is called aquerelement of g and denoted by g DORNTE [1929], such that
μn [h, g] = g, (5.23)
where g can be on any place. Any idempotent g coincides with itsquerelement g = g. It follows from (5.23) and (5.10), that the polyad
ng =(gn−2g
)(5.24)
is neutral for any element of a polyadic group, where g can be onany place. The number of relations in (5.23) can be reduced from n
(the number of possible places) to only 2 (when g is on the first andlast places DORNTE [1929], TIMM [1972], or on some other 2places ). In a polyadic group the Dornte relations
μn [g,nh;i] = μn [nh;j , g] = g (5.25)
hold true for any allowable i, j. Analog of g ∙ h ∙ h−1 = h ∙ h−1 ∙ g = g.
22
5 Special elements and properties of n-ary systems
The relation (5.23) can be treated as a definition of the unaryqueroperation
μ1 [g] = g. (5.26)
Definition 5.17. A polyadic group is a universal algebra
Ggrpn = 〈G | μn, μ1 | associativity, Dornte relations〉 , (5.27)
where μn is n-ary associative operation and μ1 is thequeroperation (5.26), such that the following diagram
G×(n)id×(n−1) ×μ1 G×n
μ1×id×(n−1)
G×n
G×G
id×Diag(n−1)Pr1
G
μnPr2
G×G
Diag(n−1)×id
(5.28)
commutes, where μ1 can be only on the first and second placesfrom the right (resp. left) on the left (resp. right) part of the diagram.
23
5 Special elements and properties of n-ary systems
A straightforward generalization of the queroperation concept and
corresponding definitions can be made by substituting in the above
formulas (5.23)–(5.26) the n-ary multiplication μn by the iterating
multiplication μ`μn (4.1) (cf. DUDEK [1980] for `μ = 2 and GAL’MAK
[2007]).
Definition 5.18. Let us define the querpower k of g recursively
g〈〈k〉〉 =(g〈〈k−1〉〉
), (5.29)
where g〈〈0〉〉 = g, g〈〈1〉〉 = g, or as the k composition
μ◦k1 =
k︷ ︸︸ ︷μ1 ◦ μ1 ◦ . . . ◦ μ1 of the queroperation (5.26).
For instance, μ◦21 = μn−3n , such that for any ternary group μ◦21 = id,
i.e. one has g = g.
24
5 Special elements and properties of n-ary systems
The negative polyadic power of an element g by (after use of (5.2))
μn
[g〈`μ−1〉, gn−2, g〈−`μ〉
]= g, μ`μn
[g`μ(n−1), g〈−`μ〉
]= g. (5.30)
Connection of the querpower and the polyadic power by the Heine
numbers HEINE [1878] or q-numbers KAC AND CHEUNG [2002]
[[k]]q =qk − 1q − 1
, (5.31)
which have the “nondeformed” limit q → 1 as [k]q → k. Then
g〈〈k〉〉 = g〈−[[k]]2−n〉, (5.32)
Assertion 5.19. The querpower coincides with the negative
polyadic deformed power with the “deformation” parameter q which
is equal to the “deviation” (2− n) from the binary group.
25
6 (One-place) homomorphisms of polyadic systems
6 (One-place) homomorphisms of polyadic systems
Let Gn = 〈G | μn〉 and G′n′ = 〈G′ | μ′n′〉 be two polyadic systems of
any kind (quasigroup, semigroup, group, etc.). If they have the
multiplications of the same arity n = n′, then one can define the
mappings from Gn to G′n. Usually such polyadic systems are
similar, and we call mappings between them the equiary mappings.
Let us take n+ 1 (one-place) mappings ϕGG′
i : G→ G′,
i = 1, . . . , n+ 1. An ordered system of mappings{ϕGG
′
i
}is called
a homotopy from Gn to G′n, if
ϕGG′
n+1 (μn [g1, . . . , gn]) = μ′n
[ϕGG
′
1 (g1) , . . . , ϕGG′
n (gn)], gi ∈ G.
(6.1)
26
6 (One-place) homomorphisms of polyadic systems
In general, one should add to this definition the “mapping” of themultiplications
μnψ(μμ′)nn′7→ μ′n′ . (6.2)
In such a way, the homotopy can be defined as the (extended)
system of mappings
{
ϕGG′
i ;ψ(μμ′)nn
}
. The corresponding
commutative (equiary) diagram is
GϕGG
′
n+1G′
...............ψ(μ)nn ................
G×n
μnϕGG
′
1 ×...×ϕGG′
n (G′)×n
μ′n (6.3)
If all the components ϕGG′
i of a homotopy are bijections, it is calledan isotopy. In case of polyadic quasigroups BELOUSOV [1972] allmappings ϕGG
′
i are usually taken as permutations of the sameunderlying set G = G′.
27
6 (One-place) homomorphisms of polyadic systems
If the multiplications are also coincide μn = μ′n, then the set
{ϕGGi ; id
}is called an autotopy of the polyadic system Gn.
The diagonal counterparts of homotopy, isotopy and autotopy
(when all mappings ϕGGi coincide) are homomorphism,
isomorphism and automorphism.
A homomorphism from Gn to G′n is given, if there exists one
mapping ϕGG′: G→ G′ satisfying
ϕGG′
(μn [g1, . . . , gn]) = μ′n
[ϕGG
′
(g1) , . . . , ϕGG′ (gn)
], gi ∈ G.
(6.4)
Usually the homomorphism is denoted by the same one letter
ϕGG′or the extended pair of mappings
{
ϕGG′;ψ(μμ′)nn
}
.
They “...are so well known that we shall not bother to define them
carefully” HOBBY AND MCKENZIE [1988].
28
7 Standard Hosszu-Gluskin theorem
7 Standard Hosszu-Gluskin theorem
Consider concrete forms of polyadic multiplication in terms oflesser arity operations.
History. Simplest way of constructing a n-ary product μ′n from thebinary one μ2 = (∗) is `μ = n iteration (4.1) SUSCHKEWITSCH
[1935], MILLER [1935]
μ′n [g] = g1 ∗ g2 ∗ . . . ∗ gn, gi ∈ G. (7.1)
In DORNTE [1929] it was noted that not all n-ary groups have aproduct of this special form.
The binary group G∗2 = 〈G | μ2 = ∗, e〉 was called a covering groupof the n-ary group G′n = 〈G | μ
′n〉 in POST [1940] (also, TVERMOES
[1953]), where a theorem establishing a more general (than (7.1))structure of μ′n [g] in terms of subgroup structure was given.
29
7 Standard Hosszu-Gluskin theorem
A manifest form of the n-ary group product μ′n [g] in terms of the
binary one and a special mapping was found in HOSSZU [1963],
GLUSKIN [1965] and is called the Hosszu-Gluskin theorem, despite
the same formulas having appeared much earlier in TURING
[1938], POST [1940] (relationship between all the formulations in
GAL’MAK AND VOROBIEV [2013]).
Rewrite (7.1) in its equivalent form
μ′n [g] = g1 ∗ g2 ∗ . . . ∗ gn ∗ e, gi, e ∈ G, (7.2)
where e is a distinguished element of the binary group 〈G | ∗, e〉,
that is the identity. Now we apply to (7.2) an “extended” version of
the homotopy relation (6.1) with Φi = ψi, i = 1, . . . n, and the l.h.s.
mapping Φn+1 = id, but add an action ψn+1 on the identity e
μn [g] = μ(e)n [g] = ψ1 (g1) ∗ ψ2 (g2) ∗ . . . ∗ ψn (gn) ∗ ψn+1 (e) . (7.3)
30
7 Standard Hosszu-Gluskin theorem
The most general form of polyadic multiplication in terms of (n+ 1)
“extended” homotopy maps ψi, i = 1, . . . n+ 1, the diagram
G×(n) × {•}id×n ×μ(e)0 G×(n+1)
ψ1×...×ψn+1G×(n+1)
G×(n)
ε
μ(e)n G
μ×n2 (7.4)
commutes.
We can correspondingly classify polyadic systems as:
1) Homotopic polyadic systems presented in the form (7.3). (7.5)
2) Nonhomotopic polyadic systems of other than (7.3) form. (7.6)
If the second class is nonempty, it would be interesting to find
examples of nonhomotopic polyadic systems.
31
7 Standard Hosszu-Gluskin theorem
The main idea in constructing the “automatically” associative n-ary
operation μn in (7.3) is to express the binary multiplication (∗) and
the “extended” homotopy maps ψi in terms of μn itself SOKOLOV
[1976]. A simplest binary multiplication which can be built from μnis (see (5.15))
g ∗t h = μn [g, t, h] , (7.7)
where t is any fixed polyad of length (n− 2). The equations for the
identity e in a binary group g ∗t e = g, e ∗t h = h, correspond to
μn [g, t, e] = g, μn [e, t, h] = h. (7.8)
We observe from (7.8) that (t, e) and (e, t) are neutral sequences
of length (n− 1), and therefore we take t as a polyadic inverse of e
(the identity of the binary group) considered as an element (but not
an identity) of the polyadic system 〈G | μn〉, so formally t = e−1.
32
7 Standard Hosszu-Gluskin theorem
Then, the binary multiplication is
g ∗ h = g ∗e h = μn[g, e−1, h
]. (7.9)
Remark 7.1. Using this construction any element of the polyadicsystem 〈G | μn〉 can be distinguished and may serve as the identityof the binary group, and is then denoted by e .
Recognize in (7.9) a version of the Maltsev term (see, e.g.,BERGMAN [2012]), which can be called a polyadic Maltsev term andis defined as
p (g, e, h)def= μn
[g, e−1, h
](7.10)
having the standard term properties p (g, e, e) = g, p (e, e, h) = h.
For n-ary group we can write g−1 =(gn−3, g
)and the binary group
inverse g−1 is g−1 = μn[e, gn−3, g, e
], the polyadic Maltsev term
becomes SHCHUCHKIN [2003]
p (g, e, h) = μn[g, en−3, e, h
]. (7.11)
33
7 Standard Hosszu-Gluskin theorem
Derive the Hosszu-Gluskin “chain formula” for ternary n = 3 case,
and then it will be clear how to proceed for generic n. We write
μ3 [g, h, u] = ψ1 (g) ∗ ψ2 (h) ∗ ψ3 (u) ∗ ψ4 (e) (7.12)
and try to construct ψi in terms of the ternary product μ3 and the
binary identity e. A neutral ternary polyad (e, e) or its powers(ek, ek
). Thus, taking for all insertions the minimal number of
neutral polyads, we get
μ3 [g, h, u] = μ73
g,
∗
↓e , e, h, e,
∗
↓e , e, e, u, e, e,
∗
↓e , e, e, e
. (7.13)
34
7 Standard Hosszu-Gluskin theorem
We rewrite (7.13) as
μ3 [g, h, u] = μ33
g,
∗
↓e , μ3 [e, h, e] ,
∗
↓e ,μ23 [e, e, u, e, e] ,
∗
↓e , μ3 [e, e, e]
.
(7.14)
Comparing this with (7.12), we can identify
ψ1 (g) = g, (7.15)
ψ2 (g) = ϕ (g) , (7.16)
ψ3 (g) = ϕ (ϕ (g)) = ϕ2 (g) , (7.17)
ψ4 (e) = μ3 [e, e, e] = e〈1〉, (7.18)
ϕ (g) = μ3 [e, g, e] . (7.19)
35
7 Standard Hosszu-Gluskin theorem
Thus, we get the Hosszu-Gluskin “chain formula” for n = 3
μ3 [g, h, u] = g ∗ ϕ (h) ∗ ϕ2 (u) ∗ b, (7.20)
b = e〈1〉. (7.21)
The polyadic power e〈1〉 is a fixed point, because ϕ(e〈1〉)= e〈1〉, as
well as higher polyadic powers e〈k〉 = μk3[e2k+1
]of the binary
identity e are obviously also fixed points ϕ(e〈k〉
)= e〈k〉.
By analogy, the Hosszu-Gluskin “chain formula” for arbitrary n can
be obtained using substitution e→ e−1, neutral polyads(e−1, e
)
and their powers((e−1
)k, ek)
, the mapping ϕ in the n-ary case is
ϕ (g) = μn[e, g, e−1
], (7.22)
and μn [e, . . . , e] is also the first n-ary power e〈1〉 (5.2).
36
7 Standard Hosszu-Gluskin theorem
In this way, we obtain the Hosszu-Gluskin “chain formula” for
arbitrary n
μn [g1, . . . , gn] = g1∗ϕ (g2)∗ϕ2 (g3)∗. . .∗ϕ
n−2 (gn−1)∗ϕn−1 (gn)∗e
〈1〉.
(7.23)
Thus, we have found the “extended” homotopy maps ψi from (7.3)
ψi (g) = ϕi−1 (g) , i = 1, . . . , n, (7.24)
ψn+1 (g) = g〈1〉, (7.25)
where by definition ϕ0 (g) = g. Using (7) and (7.23) we can
formulate the standard Hosszu-Gluskin theorem in the language of
polyadic powers.
Theorem 7.2. On a polyadic group Gn = 〈G | μn, μ1〉 one can
define a binary group G∗2 = 〈G | μ2 = ∗, e〉 and its automorphism ϕ
such that the Hosszu-Gluskin “chain formula” (7.23) is valid.
37
7 Standard Hosszu-Gluskin theorem
The following reverse Hosszu-Gluskin theorem holds.
Theorem 7.3. If in a binary group G∗2 = 〈G | μ2 = ∗, e〉 one can
define an automorphism ϕ such that
ϕn−1 (g) = b ∗ g ∗ b−1, (7.26)
ϕ (b) = b, (7.27)
where b ∈ G is a distinguished element, then the “chain formula”
μn [g1, . . . , gn] = g1 ∗ϕ (g2)∗ϕ2 (g3)∗ . . .∗ϕ
n−2 (gn−1)∗ϕn−1 (gn)∗ b.
(7.28)
determines a n-ary group, in which the distinguished element is the
first polyadic power of the binary identity b = e〈1〉.
38
8 “Deformation” of Hosszu-Gluskin chain formula
8 “Deformation” of Hosszu-Gluskin chain formula
We can generalize the Hosszu-Gluskin chain formula, if the number
of the inserted neutral polyads can be chosen arbitrarily, not only
minimally, as they are neutral. Indeed, in the particular case n = 3,
we put the map ϕ as
ϕq (g) = μ`ϕ(q)3 [e, g, eq] , (8.1)
where the number of multiplications
`ϕ (q) =q + 1
2(8.2)
is an integer `ϕ (q) = 1, 2, 3 . . ., while q = 1, 3, 5, 7 . . .. Then, we get
μ3 [g, h, u] = μ•3
[g, e, (e, h, eq) , e, (e, u, eq)
q+1, e, eq(q+1)+1
]. (8.3)
39
8 “Deformation” of Hosszu-Gluskin chain formula
Therefore we have obtained the “q-deformed” homotopy maps
ψ1 (g) = ϕ[[0]]qq (g) = ϕ0q (g) = g, (8.4)
ψ2 (g) = ϕq (g) = ϕ[[1]]qq (g) , (8.5)
ψ3 (g) = ϕq+1q (g) = ϕ
[[2]]qq (g) , (8.6)
ψ4 (g) = μ•3
[gq(q+1)+1
]= μ•3
[g[[3]]q
], (8.7)
where ϕ is defined by (8.1) and [[k]]q is the q-deformed number andwe put ϕ0q = id. The corresponding “q-deformed” chain formula (forn = 3) can be written as
μ3 [g, h, u] = g ∗ ϕ[[1]]qq (h) ∗ ϕ
[[2]]qq (u) ∗ bq, (8.8)
bq = e〈`e(q)〉, (8.9)
`e (q) = q[[2]]q2
. (8.10)
40
8 “Deformation” of Hosszu-Gluskin chain formula
The “nondeformed” limit q → 1 of (8.8) gives the standard
Hosszu-Gluskin chain formula (7.20) for n = 3. For arbitrary n we
insert all possible powers of neutral polyads((e−1
)k, ek)
(they are
allowed by the chain properties), and obtain
ϕq (g) = μ`ϕ(q)n
[e, g,
(e−1
)q], (8.11)
where the number of multiplications `ϕ (q) =q (n− 2) + 1
n− 1is an
integer and `ϕ (q)→ q, as n→∞, and `ϕ (1) = 1, as in (7.22).
41
8 “Deformation” of Hosszu-Gluskin chain formula
The “deformed” map ϕq is a kind of a-quasi-endomorphismGLUSKIN AND SHVARTS [1972] (which has one multiplication andleads to the standard “nondeformed” chain formula) of the binarygroup G∗2, because from (8.11) we get
ϕq (g) ∗ ϕq (h) = ϕq (g ∗ a ∗ h) , (8.12)
where a = ϕq (e). A general quasi-endomorphism DUPLIJ [2013b]
ϕq (g) ∗ ϕq (h) = ϕq(g ∗ ϕq (e) ∗ h
). (8.13)
The corresponding diagram
G×Gμ2 G
ϕqG
G×G
ϕq×ϕqεG× {•} ×G
id×μ(e)0 ×id G×G×G
μ2×μ2 (8.14)
commutes. If q = 1, then ϕq (e) = e, and the distinguished elementa turns to binary identity a = e, and ϕq is an automorphism of G∗2.
42
8 “Deformation” of Hosszu-Gluskin chain formula
The “extended” homotopy maps ψi (7.3) now are
ψ1 (g) = ϕ[[0]]qq (g) = ϕ0q (g) = g, (8.15)
ψ2 (g) = ϕq (g) = ϕ[[1]]qq (g) , (8.16)
ψ3 (g) = ϕq+1q (g) = ϕ
[[2]]qq (g) , (8.17)
...
ψn−1 (g) = ϕqn−3+...+q+1q (g) = ϕ
[[n−2]]qq (g) , (8.18)
ψn (g) = ϕqn−2+...+q+1q (g) = ϕ
[[n−1]]qq (g) , (8.19)
ψn+1 (g) = μ•n
[gqn−1+...+q+1
]= μ•n
[g[[n]]q
]. (8.20)
In terms of the polyadic power (5.2), the last map is
ψn+1 (g) = g〈`e〉, `e (q) = q
[[n− 1]]qn− 1
. (8.21)
43
8 “Deformation” of Hosszu-Gluskin chain formula
Thus the “q-deformed” n-ary chain formula is DUPLIJ [2013b]
μn [g1, . . . , gn] = g1∗ϕ[[1]]qq (g2)∗ϕ
[[2]]qq (g3)∗. . .∗ϕ
[[n−2]]qq (gn−1)∗ϕ
[[n−1]]qq (gn)∗e
〈`e(q)〉.
(8.22)
In the “nondeformed” limit q → 1 (8.22) reproduces the standardHosszu-Gluskin chain formula (7.23). Instead of the fixed pointrelation (7.27) we now have the quasi-fixed point
ϕq (bq) = bq ∗ ϕq (e) , (8.23)
where the “deformed” distinguished element bq is
bq = μ•n
[e[[n]]q
]= e〈`e(q)〉. (8.24)
The conjugation relation (7.26) in the “deformed” case becomesthe quasi-conjugation DUPLIJ [2013b]
ϕ[[n−1]]qq (g) ∗ bq = bq ∗ ϕ
[[n−1]]qq (e) ∗ g. (8.25)
44
8 “Deformation” of Hosszu-Gluskin chain formula
We formulate the following “q-deformed” analog of the
Hosszu-Gluskin theorem DUPLIJ [2013b].
Theorem 8.1. On a polyadic group Gn = 〈G | μn, μ1〉 one can
define a binary group G∗2 = 〈G | μ2 = ∗, e〉 and (the infinite
“q-series” of) its automorphism ϕq such that the “deformed” chain
formula (8.22) is valid
μn [g1, . . . , gn] =
(
∗n∏
i=1
ϕ[[i−1]]q (gi)
)
∗ bq, (8.26)
where (the infinite “q-series” of) the “deformed” distinguished
element bq (being a polyadic power of the binary identity (8.24)) is
the quasi-fixed point of ϕq (8.23) and satisfies the
quasi-conjugation (8.25) in the form
ϕ[[n−1]]qq (g) = bq ∗ ϕ
[[n−1]]qq (e) ∗ g ∗ b−1q . (8.27)
45
9 (One-place) generalizations of homomorphisms
9 (One-place) generalizations of homomorphisms
Definition 9.1. The n-ary homomorphism is realized as asequence of n consequent (binary) homomorphisms ϕi,i = 1, . . . , n, of n similar polyadic systems
n︷ ︸︸ ︷
Gnϕ1→ G′n
ϕ2→ . . .ϕn−1→ G′′n
ϕn→ G′′′n (9.1)
Generalized POST [1940] n-adic substitutions in GAL’MAK [1998].
There are two possibilities to change arity:
1) add another equiary diagram with additional operations using thesame formula (6.4), where both do not change arity (are equiary);
2) use one modified (and not equiary) diagram and the underlyingformula (6.4) by themselves, which will allow us to change aritywithout introducing additional operations.
46
9 (One-place) generalizations of homomorphisms
The first way leads to the concept of weak homomorphism which
was introduced in GOETZ [1966], MARCZEWSKI [1966], GŁAZEK
AND MICHALSKI [1974] for non-indexed algebras and in GŁAZEK
[1980] for indexed algebras, then developed in TRACZYK [1965] for
Boolean and Post algebras, in DENECKE AND WISMATH [2009] for
coalgebras and F -algebras DENECKE AND SAENGSURA [2008].
Incorporate into the polyadic systems 〈G | μn〉 and 〈G′ | μ′n′〉 the
following additional term operations of opposite arity
νn′ : G×n′ → G and ν′n : G
′×n → G′ and consider two equiary
mappings between 〈G | μn, νn′〉 and 〈G′ | μ′n′ , ν′n〉.
47
9 (One-place) generalizations of homomorphisms
A weak homomorphism from 〈G | μn, νn′〉 to 〈G′ | μ′n′ , ν′n〉 is given,
if there exists a mapping ϕGG′: G→ G′ satisfying two relations
simultaneously
ϕGG′
(μn [g1, . . . , gn]) = ν′n
[ϕGG
′
(g1) , . . . , ϕGG′ (gn)
], (9.2)
ϕGG′
(νn′ [g1, . . . , gn′ ]) = μ′n′
[ϕGG
′
(g1) , . . . , ϕGG′ (gn′)
]. (9.3)
GϕGG
′
G′
........ψ(μν′)
nn.........
G×n
μn (ϕGG
′)×n
(G′)×n
ν′n
GϕGG
′
G′
..........ψ(νμ′)
n′n′...........
G×n′
νn′(ϕGG
′)×n′
(G′)×n′
μ′n′ (9.4)
If only one of the relations (9.2) or (9.3) holds, such a mapping is
called a semi-weak homomorphism KOLIBIAR [1984]. If ϕGG′
is
bijective, then it defines a weak isomorphism.
48
10 Multiplace mappings and heteromorphisms
10 Multiplace mappings and heteromorphisms
Second way of changing the arity: use only one relation (diagram).
Idea. Using the additional distinguished mapping: the identity idG.
Define an (`id-intact) id-product for the n-ary system 〈G | μn〉 as
μ(`id)n = μn × (idG)×`id , (10.1)
μ(`id)n : G×(n+`id) → G×(1+`id). (10.2)
To indicate the exact i-th place of μn in (10.1), we write μ(`id)n (i).
Introduce a multiplace mapping Φ(n,n′)k acting as DUPLIJ [2013a]
Φ(n,n′)k : G×k → G′. (10.3)
49
10 Multiplace mappings and heteromorphisms
We have the following commutative diagram which changes arity
G×kΦk
G′
G×kn
μ(`id)n
(Φk)×n′
(G′)×n′μ′n′ (10.4)
Definition 10.1. A k-place heteromorphism from Gn to G′n′ is
given, if there exists a k-place mapping Φ(n,n′)k (10.3) such that the
corresponding defining equation (a modification of (6.4)) depends
on the place i of μn in (10.1).
50
10 Multiplace mappings and heteromorphisms
For i = 1 it can read as DUPLIJ [2013a]
Φ(n,n′)k
μn [g1, . . . , gn]
gn+1...
gn+`id
= μ′n′
Φ(n,n′)k
g1...
gk
, . . . ,Φ
(n,n′)k
gk(n′−1)...
gkn′
.
(10.5)
The notion of heteromorhism is motivated by ELLERMAN [2006,
2007], where mappings between objects from different categories
were considered and called ’chimera morphisms’.
51
10 Multiplace mappings and heteromorphisms
In the particular case n = 3, n′ = 2, k = 2, `id = 1 we have
Φ(3,2)2
μ3 [g1, g2, g3]
g4
= μ′2
Φ(3,2)2
g1
g2
,Φ(3,2)2
g3
g4
.
(10.6)
This was used in the construction of the bi-element representations
of ternary groups BOROWIEC ET AL. [2006], DUPLIJ [2013a].
Example 10.2. Let G =Madiag2 (K), a set of antidiagonal 2× 2
matrices over the field K and G′ = K, where K = R,C,Q,H. The
ternary multiplication μ3 is a product of 3 matrices. Obviously, μ3 is
nonderived.
52
10 Multiplace mappings and heteromorphisms
For the elements gi =
0 ai
bi 0
, i = 1, 2, we construct a 2-place
mapping G×G→ G′ as
Φ(3,2)2
g1
g2
= a1 a2b1 b2, (10.7)
which satisfies (10.6). Introduce a 1-place mapping by ϕ (gi) = aibi,which satisfies the standard (6.4) for a commutative field K only(= R,C) becoming a homomorphism. So we can have the relationbetween the heteromorhism Φ(3,2)2 and the homomorphism ϕ
Φ(3,2)2
g1
g2
= ϕ (g1) ∙ ϕ (g2) = a1 b1a2 b2, (10.8)
where the product (∙) is in K, such that (6.4) and (10.6) coincide.
53
10 Multiplace mappings and heteromorphisms
For the noncommutative field K (= Q or H) we can define the
heteromorphism (10.7) only.
A heteromorphism is called derived, if it can be expressed through
an ordinary (one-place) homomorphism (as e.g., (10.8)).
A heteromorphism is called a `μ-ple heteromorphism, if it contains
`μ multiplications in the argument of Φ(n,n′)k in its defining relation.
We define a `μ-ple `id-intact id-product for 〈G;μn〉 as
μ(`μ,`id)n = (μn)×`μ × (idG)
×`id , (10.9)
μ(`μ,`id)n : G×(n`μ+`id) → G×(`μ+`id). (10.10)
A `μ-ple k-place heteromorphism from Gn to G′n′ is given, if there
exists a k-place mapping Φ(n,n′)k (10.3).
54
10 Multiplace mappings and heteromorphisms
The main heteromorphism equation is DUPLIJ [2013a]
Φ
(n,n′
)
k
μn [g1, . . . , gn] ,
.
.
.
μn
[gn(`μ−1
), . . . , gn`μ
]
`μ
gn`μ+1,
.
.
.
gn`μ+`id
`id
= μ′n′
Φ
(n,n′
)
k
g1
.
.
.
gk
, . . . ,Φ
(n,n′
)
k
gk(n′−1
)
.
.
.
gkn′
.
(10.11)
It is a polyadic analog of Φ(g1 ∗ g2) = Φ (g1) • Φ(g2), whichcorresponds to n = 2, n′ = 2, k = 1, `μ = 1, `id = 0, μ2 = ∗, μ
′2 = •.
We obtain two arity changing formulas
n′ = n−n− 1k
`id, (10.12)
n′ =n− 1k
`μ + 1, (10.13)
where n−1k`id ≥ 1 and n−1
k`μ ≥ 1 are integer.
55
10 Multiplace mappings and heteromorphisms
The following inequalities hold valid
1 ≤ `μ ≤ k, (10.14)
0 ≤ `id ≤ k − 1, (10.15)
`μ ≤ k ≤ (n− 1) `μ, (10.16)
2 ≤ n′ ≤ n. (10.17)
The main statement follows from (10.17):
The heteromorphism Φ(n,n′)k decreases arity of the multiplication.
If `id 6= 0 then it is change of the arity n′ 6= n.
If `id = 0, then k = kmin = `μ, and no change of arity n′max = n.
We call a heteromorphism having `id = 0 a k-place homomorphismwith k = `μ. An opposite extreme case, when the final arityapproaches its minimum n′min = 2 (the final operation is binary),corresponds to the maximal value of places k = kmax = (n− 1) `μ.
56
10 Multiplace mappings and heteromorphisms
Рис. 1:
Dependence of the final arity n′ through the number of heteromorphism
places k for the fixed initial arity n = 9 with
left: fixed intact elements `id = const (`id = 1 (solid), `id = 2 (dash));
right: fixed multiplications `μ = const (`μ = 1 (solid), `μ = 2 (dash)).
57
10 Multiplace mappings and heteromorphisms
Theorem 10.3. Any n-ary system can be mapped into a binary
system by binarizing heteromorphism Φ(n,2)(n−1)`μ, `id = (n− 2) `μ.
Proposition 10.4. Classification of `μ-ple heteromorphisms:
1. n′ = n′max = n =⇒ Φ(n,n)`μ
is the `μ-place homomorphism,
k = kmin = `μ.
2. 2 < n′ < n =⇒ Φ(n,n′)k is the intermediate heteromorphism with
k =n− 1n′ − 1
`μ, `id =n− n′
n′ − 1`μ. (10.18)
3. n′ = n′min = 2 =⇒ Φ(n,2)(n−1)`μ
is the (n− 1) `μ-place binarizing
heteromorphism, i.e., k = kmax = (n− 1) `μ.
58
10 Multiplace mappings and heteromorphisms
Таблица 1: “Quantization” of heteromorphisms
k `μ `id n/n′
2 1 1n = 3, 5, 7, . . .
n′ = 2, 3, 4, . . .
3 1 2n = 4, 7, 10, . . .
n′ = 2, 3, 4, . . .
3 2 1n = 4, 7, 10, . . .
n′ = 3, 5, 7, . . .
4 1 3n = 5, 9, 13, . . .
n′ = 2, 3, 4, . . .
4 2 2n = 3, 5, 7, . . .
n′ = 2, 3, 4, . . .
4 3 1n = 5, 9, 13, . . .
n′ = 4, 7, 10, . . .59
11 Associativity, quivers and heteromorphisms
11 Associativity, quivers and heteromorphisms
Semigroup heteromorphisms: associativity of the final operation
μ′n′ , when the initial operation μn is associative.
A polyadic quiver of product is the set of elements from Gn and
arrows, such that the elements along arrows form n-ary product μnDUPLIJ [2013a]. For instance, for the multiplication μ4 [g1, h2, g2, u1]
the 4-adic quiver is denoted by {g1 → h2 → g2 → u1}.
Define polyadic quivers which are related to the main
heteromorphism equation (10.11).
60
11 Associativity, quivers and heteromorphisms
For example, the polyadic quiver {g1 → h2 → g2 → u1;h1, u2}
corresponds to the heteromorphism with n = 4, n′ = 2, k = 3,
`id = 2 and `μ = 1 is
Φ(4,2)3
μ4 [g1, h2, g2, u1]
h1
u2
= μ
′2
Φ(4,2)3
g1
h1
u1
,Φ
(4,2)3
g2
h2
u2
.
(11.1)
As it is seen from (11.1), the product μ′2 is not associative, if μ4 is
associative.
Definition 11.1. An associative polyadic quiver is a polyadic quiver
which ensures the final associativity of μ′n′ in the main
heteromorphism equation (10.11), when the initial multiplication μnis associative.
61
11 Associativity, quivers and heteromorphisms
One of the associative polyadic quivers which corresponds to thesame heteromorphism parameters as the non-associative quiver(11.1) is {g1 → h2 → u1 → g2;h1, u2} which corresponds to
g1 h1 u1 g1 h1 u1
g2 h2 u2 g2 h2 u2
corr
Φ(4,2)3
μ4 [g1, h2, u1, g2]
h1
u2
= μ′2
Φ
(4,2)3
g1
h1
u1
,Φ
(4,2)3
g2
h2
u2
.
(11.2)
We propose a classification of associative polyadic quivers and therules of construction of corresponding heteromorphism equations,i.e. consistent procedure for building semigroup heteromorphisms.
62
11 Associativity, quivers and heteromorphisms
The first class of heteromorphisms (`id = 0 or intactless), that is
`μ-place (multiplace) homomorphisms. As an example, for
n = n′ = 3, k = 2, `μ = 2 we have
Φ(3,3)2
μ3 [g1, g2, g3]
μ3 [h1, h2, h3]
= μ′3
Φ(3,3)2
g1
h1
,Φ(3,3)2
g2
h2
,Φ(3,3)2
g3
h3
.
(11.3)
Note that the analogous quiver with opposite arrow directions is
Φ(3,3)2
μ3 [g1, g2, g3]
μ3 [h3, h2, h1]
= μ′3
Φ(3,3)2
g1
h1
,Φ(3,3)2
g2
h2
,Φ(3,3)2
g3
h3
.
(11.4)
It was used in constructing the middle representations of ternary
groups BOROWIEC ET AL. [2006], DUPLIJ [2013a].
63
11 Associativity, quivers and heteromorphisms
An important class of intactless heteromorphisms (with `id = 0)preserving associativity can be constructed using an analogy withthe Post substitutions POST [1940], and therefore we call it thePost-like associative quiver. The number of places k is now fixed byk = n− 1, while n′ = n and `μ = k = n− 1. An example of thePost-like associative quiver with the same heteromorphismsparameters as in (11.3)-(11.4) is
Φ(3,3)2
μ3 [g1, h2, g3]
μ3 [h1, g2, h3]
= μ′3
Φ(3,3)2
g1
h1
,Φ(3,3)2
g2
h2
,Φ(3,3)2
g3
h3
.
(11.5)This construction appeared in the study of ternary semigroups ofmorphisms CHRONOWSKI [1994]. Its n-ary generalization was usedspecial representations of n-groups GLEICHGEWICHT ET AL. [1983],WANKE-JAKUBOWSKA AND WANKE-JERIE [1984] (where the n-groupwith the multiplication μ′2 was called the diagonal n-group).
64
12 Multiplace representations of polyadic systems
12 Multiplace representations of polyadic systems
The final multiplication μ′n′ is a linear map, which leads to
restrictions on the final polyadic structure G′n′ .
Let V be a vector space over a field K and EndV be a set of linear
endomorphisms of V , which is in fact a binary group. In the
standard way, a linear representation of a binary semigroup
G2 = 〈G;μ2〉 is a (one-place) map Π1 : G2 → EndV , such that Π1is a (one-place) homomorphism
Π1 (μ2 [g, h]) = Π1 (g) ∗Π1 (h) , (12.1)
where g, h ∈ G and (∗) is the binary multiplication in EndV .
65
12 Multiplace representations of polyadic systems
If G2 is a binary group with the unity e, then we have the additional
condition
Π1 (e) = idV . (12.2)
General idea: to use the heteromorphism equation (10.11) instead
of the standard homomorphism equation (12.1), such that the arity
of the representation will be different from the arity of the initial
polyadic system n′ 6= n.
Consider the structure of the final n′-ary multiplication μ′n′ in
(10.11), taking into account that the final polyadic system G′n′should be constructed from EndV . The most natural and physically
applicable way is to consider the binary EndV and to put
G′n′ = dern′ (EndV ), as it was proposed for the ternary case in
BOROWIEC ET AL. [2006], DUPLIJ [2013a].
66
12 Multiplace representations of polyadic systems
In this way G′n′ becomes a derived n′-ary (semi)group of
endomorphisms of V with the multiplication μ′n′ :
(EndV )×n′ → EndV , where
μ′n′ [v1, . . . , vn′ ] = v1 ∗ . . . ∗ vn′ , vi ∈ EndV. (12.3)
Because the multiplication μ′n′ (12.3) is derived and is therefore
associative, consider the associative initial polyadic systems.
Let Gn = 〈G | μn〉 be an associative n-ary polyadic system. By
analogy with (10.3), we introduce the following k-place mapping
Π(n,n′)k : G×k → EndV. (12.4)
67
12 Multiplace representations of polyadic systems
A multiplace representation of an associative polyadic system Gn
in a vector space V is given, if there exists a k-place mapping(12.4) which satisfies the (associativity preserving)heteromorphism equation (10.11), that is DUPLIJ [2013a]
Π
(n,n′
)
k
μn [g1, . . . , gn] ,
.
.
.
μn
[gn(`μ−1
), . . . , gn`μ
]
`μ
gn`μ+1,
.
.
.
gn`μ+`id
`id
=
n′︷ ︸︸ ︷
Π
(n,n′
)
k
g1
.
.
.
gk
∗ . . . ∗ Π
(n,n′
)
k
gk(n′−1
)
.
.
.
gkn′
,
(12.5)
G×kΠk
EndV
G×kn′
μ(`μ,`id)n
(Πk)×n′
(EndV )×n′
(∗)n′ (12.6)
where μ(`μ,`id)n is `μ-ple `id-intact id-product given by (10.9).
68
12 Multiplace representations of polyadic systems
General classification of multiplace representations can be done byanalogy with that of the heteromorphisms as follows:
1. The hom-like multiplace representation which is a multiplacehomomorphism with n′ = n′max = n, without intact elements
lid = l(min)id = 0, and minimal number of places k = kmin = `μ.
2. The intact element multiplace representation which is theintermediate heteromorphism with 2 < n′ < n and the numberof intact elements is
lid =n− n′
n′ − 1`μ. (12.7)
3. The binary multiplace representation which is a binarizingheteromorphism (3) with n′ = n′min = 2, the maximal number of
intact elements l(max)id = (n− 2) `μ and maximal number ofplaces
k = kmax = (n− 1) `μ. (12.8)
69
12 Multiplace representations of polyadic systems
In case of n-ary groups, we need an analog of the “normalizing”
relation (12.2). If the n-ary group has the unity e, then
Π(n,n′)k
e
...
e
k
= idV . (12.9)
70
12 Multiplace representations of polyadic systems
If there is no unity at all, one can “normalize” the multiplacerepresentation, using analogy with (12.2) in the form
Π1(h−1 ∗ h
)= idV , (12.10)
as follows
Π(n,n′)k
h
...
h
`μ
h
...
h
`id
= idV , (12.11)
for all h ∈Gn, where h is the querelement of h. The latter ones canbe placed on any places in the l.h.s. of (12.11) due to the Dornteidentities.
71
12 Multiplace representations of polyadic systems
A general form of multiplace representations can be found by
applying the Dornte identities to each n-ary product in the l.h.s. of
(12.5). Then, using (12.11) we have schematically
Π(n,n′)k
g1...
gk
= Π(n,n′)k
t1...
t`μ
g
...
g
`id
, (12.12)
where g is an arbitrary fixed element of the n-ary group and
ta = μn [ga1, . . . , gan−1, g] , a = 1, . . . , `μ. (12.13)
72
13 Multiactions and G-spaces
13 Multiactions and G-spaces
Let Gn = 〈G | μn〉 be a polyadic system and X be a set. A (left)1-place action of Gn on X is the external binary operationρ(n)1 : G× X→ X such that it is consistent with the multiplicationμn, i.e. composition of the binary operations ρ1 {g|x} gives then-ary product, that is,
ρ(n)1 {μn [g1, . . . gn] |x} = ρ
(n)1
{g1|ρ
(n)1
{g2| . . . |ρ
(n)1 {gn|x}
}. . .}.
(13.1)If the polyadic system is a n-ary group, then in addition to (13.1) itis implied the there exist such ex ∈ G (which may or may notcoincide with the unity of Gn) that ρ(n)1 {ex|x} = x for all x ∈ X, and
the mapping x 7→ ρ(n)1 {ex|x} is a bijection of X. The right 1-placeactions of Gn on X are defined in a symmetric way, and thereforewe will consider below only one of them.
73
13 Multiactions and G-spaces
Obviously, we cannot compose ρ(n)1 and ρ(n′)
1 with n 6= n′. Usually
X is called a G-set or G-space depending on its properties (see,
e.g., HUSEMOLLER ET AL. [2008]).
We introduce the multiplace concept of action for polyadic systems,
which is consistent with heteromorphisms and multiplace
representations.
For a polyadic system Gn = 〈G | μn〉 and a set X we introduce an
external polyadic operation
ρk : G×k × X→ X, (13.2)
which is called a (left) k-place action or multiaction. We use the
analogy with multiplication laws of the heteromorphisms (10.11) .
and the multiplace representations (12.5).
74
13 Multiactions and G-spaces
We propose (schematically) DUPLIJ [2013a]
ρ(n)k
μn [g1, . . . , gn] ,
.
.
.
μn
[gn(`μ−1
), . . . , gn`μ
]
`μ
gn`μ+1,
.
.
.
gn`μ+`id
`id
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
x
= ρ(n)k
n′︷ ︸︸ ︷
g1
.
.
.
gk
∣∣∣∣∣∣∣∣∣∣∣
. . .
∣∣∣∣∣∣∣∣
ρ(n)k
gk(n′−1
)
.
.
.
gkn′
∣∣∣∣∣∣∣∣∣∣∣
x
. . .
.
(13.3)
The connection between all the parameters here is the same as in
the arity changing formulas (10.12)–(10.13). Composition of
mappings is associative, and therefore in concrete cases we can
use our associative quiver technique from Section 11.
75
13 Multiactions and G-spaces
If Gn is n-ary group, then we should add to (13.3) the “normalizing”
relations. If there is a unity e ∈Gn, then
ρ(n)k
e
...
e
∣∣∣∣∣∣∣∣∣
x
= x, for all x ∈ X. (13.4)
76
13 Multiactions and G-spaces
In terms of the querelement, the normalization has the form
ρ(n)k
h
...
h
`μ
h
...
h
`id
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
x
= x, for all x ∈ X and for all h ∈ Gn.
(13.5)
77
13 Multiactions and G-spaces
The multiaction ρ(n)k is transitive, if any two points x and y in X can
be “connected” by ρ(n)k , i.e. there exist g1, . . . , gk ∈Gn such that
ρ(n)k
g1...
gk
∣∣∣∣∣∣∣∣∣
x
= y. (13.6)
If g1, . . . , gk are unique, then ρ(n)k is sharply transitive. The subset
of X, in which any points are connected by (13.6) with fixed
g1, . . . , gk can be called the multiorbit of X. If there is only one
multiorbit, then we call X the heterogenous G-space (by analogy
with the homogeneous one).
78
13 Multiactions and G-spaces
By analogy with the (ordinary) 1-place actions, we define a
G-equivariant map Ψ between two G-sets X and Y by (in our
notation)
Ψ
ρ(n)k
g1...
gk
∣∣∣∣∣∣∣∣∣
x
= ρ
(n)k
g1...
gk
∣∣∣∣∣∣∣∣∣
Ψ(x)
∈ Y, (13.7)
which makes G-space into a category (for details, see, e.g.,
HUSEMOLLER ET AL. [2008]). In the particular case, when X is a
vector space over K, the multiaction (13.2) can be called a
multi-G-module which satisfies (13.4).
79
13 Multiactions and G-spaces
The additional (linearity) conditions are
ρ(n)k
g1...
gk
∣∣∣∣∣∣∣∣∣
ax+ by
= aρ(n)k
g1...
gk
∣∣∣∣∣∣∣∣∣
x
+ bρ(n)k
g1...
gk
∣∣∣∣∣∣∣∣∣
y
,
(13.8)
where a, b ∈ K. Then, comparing (12.5) and (13.3) we can define a
multiplace representation as a multi-G-module by the following
formula
Π(n,n′)k
g1...
gk
(x) = ρ
(n)k
g1...
gk
∣∣∣∣∣∣∣∣∣
x
. (13.9)
In a similar way, one can generalize to polyadic systems other
notions from group action theory, see e.g. KIRILLOV [1976].
80
14 Regular multiactions
14 Regular multiactions
The most important role in the study of polyadic systems is played
by the case, when X =Gn, and the multiaction coincides with the
n-ary analog of translations MAL’TCEV [1954], so called
i-translations BELOUSOV [1972]. In the binary case, ordinary
translations lead to regular representations KIRILLOV [1976], and
therefore we call such an action a regular multiaction ρreg(n)k . In this
connection, the analog of the Cayley theorem for n-ary groups was
obtained in GAL’MAK [1986, 2001]. Now we will show in examples,
how the regular multiactions can arise from i-translations.
81
14 Regular multiactions
Example 14.1. Let G3 be a ternary semigroup, k = 2, and X =G3,
then 2-place (left) action can be defined as
ρreg(3)2
g
h
∣∣∣∣∣∣u
def= μ3 [g, h, u] . (14.1)
This gives the following composition law for two regular multiactions
ρreg(3)2
g1
h1
∣∣∣∣∣∣ρreg(3)2
g2
h2
∣∣∣∣∣∣u
= μ3 [g1, h1, μ3 [g2, h2, u]]
= μ3 [μ3 [g1, h1, g2] , h2, u] = ρreg(3)2
μ3 [g1, h1, g2]
h2
∣∣∣∣∣∣u
. (14.2)
Thus, using the regular 2-action (14.1) we have, in fact, found the
associative quiver corresponding to (10.6).
82
14 Regular multiactions
The formula (14.1) can be simultaneously treated as a 2-translation
BELOUSOV [1972]. In this way, the following left regular multiaction
ρreg(n)k
g1...
gk
∣∣∣∣∣∣∣∣∣
h
def= μn [g1, . . . , gk, h] , (14.3)
where in the r.h.s. there is the i-translation with i = n. The right
regular multiaction corresponds to the i-translation with i = 1. In
general, the value of i fixes the minimal final arity n′reg, which
differs for even and odd values of the initial arity n.
83
14 Regular multiactions
It follows from (14.3) that for regular multiactions the number of
places is fixed
kreg = n− 1, (14.4)
and the arity changing formulas (10.12)–(10.13) become
n′reg = n− `id (14.5)
n′reg = `μ + 1. (14.6)
From (14.5)–(14.6) we conclude that for any n a regular multiaction
having one multiplication `μ = 1 is binarizing and has n− 2 intact
elements. For n = 3 see (14.2). Also, it follows from (14.5) that for
regular multiactions the number of intact elements gives exactly the
difference between initial and final arities.
84
14 Regular multiactions
If the initial arity is odd, then there exists a special middle regular
multiaction generated by the i-translation with i = (n+ 1)�2. For
n = 3 the corresponding associative quiver is (11.4) and such
2-actions were used in BOROWIEC ET AL. [2006], DUPLIJ [2013a] to
construct middle representations of ternary groups, which did not
change arity (n′ = n). Here we give a more complicated example of
a middle regular multiaction, which can contain intact elements and
can therefore change arity.
85
14 Regular multiactions
Example 14.2. Let us consider 5-ary semigroup and the following
middle 4-action
ρreg(5)4
g
h
u
v
∣∣∣∣∣∣∣∣∣∣∣
s
= μ5
g, h,
i=3↓s , u, v
. (14.7)
Using (14.6) we observe that there are two possibilities for the
number of multiplications `μ = 2, 4. The last case `μ = 4 is similar
to the vertical associative quiver (11.4), but with a more
86
14 Regular multiactions
complicated l.h.s., that is
ρreg(5)4
μ5 [g1, h1, g2, h2,g3]
μ5 [h3, g4, h4, g5, h5]
μ5 [u5, v5, u4, v4, u3]
μ5 [v3, u2, v2, u1, v1]
∣∣∣∣∣∣∣∣∣∣∣
s
=
ρreg(5)4
g1
h1
u1
v1
∣∣∣∣∣∣∣∣∣∣∣
ρreg(5)4
g2
h2
u2
v2
∣∣∣∣∣∣∣∣∣∣∣
ρreg(5)4
g3
h3
u3
v3
∣∣∣∣∣∣∣∣∣∣∣
ρreg(5)4
g4
h4
u4
v4
∣∣∣∣∣∣∣∣∣∣∣
ρreg(5)4
g5
h5
u5
v5
∣∣∣∣∣∣∣∣∣∣∣
s
.
(14.8)
87
14 Regular multiactions
Now we have an additional case with two intact elements `id and
two multiplications `μ = 2 as
ρreg(5)4
μ5 [g1, h1, g2, h2,g3]
h3
μ5 [h3, v3, u2, v2, u1]
v1
∣∣∣∣∣∣∣∣∣∣∣
s
= ρreg(5)4
g1
h1
u1
v1
∣∣∣∣∣∣∣∣∣∣∣
ρreg(5)4
g2
h2
u2
v2
∣∣∣∣∣∣∣∣∣∣∣
ρreg(5)4
g3
h3
u3
v3
∣∣∣∣∣∣∣∣∣∣∣
s
,
(14.9)
with arity changing from n = 5 to n′reg = 3. In addition to (14.9) we
have 3 more possible regular multiactions due to the associativity
of μ5, when the multiplication brackets in the sequences of 6
elements in the first two rows and the second two ones can be
shifted independently.
88
14 Regular multiactions
For n > 3, in addition to left, right and middle multiactions, there
exist intermediate cases. First, observe that the i-translations with
i = 2 and i = n− 1 immediately fix the final arity n′reg = n.
Therefore, the composition of multiactions will be similar to (14.8),
but with some permutations in the l.h.s.
Now we consider some multiplace analogs of regular
representations of binary groups KIRILLOV [1976]. The
straightforward generalization is to consider the previously
introduced regular multiactions (14.3) in the r.h.s. of (13.9). Let Gn
be a finite polyadic associative system and KGn be a vector space
spanned by Gn (some properties of n-ary group rings were
considered in ZEKOVIC AND ARTAMONOV [1999, 2002]).
89
14 Regular multiactions
This means that any element of KGn can be uniquely presented in
the form w =∑l al ∙ hl, al ∈ K, hl ∈ G. Then, using (14.3), we
define the i-regular k-place representation DUPLIJ [2013a]
Πreg(i)k
g1...
gk
(w) =
∑
l
al ∙μk+1 [g1 . . . gi−1hlgi+1 . . . gk] . (14.10)
Comparing (14.3) and (14.10) one can conclude that general
properties of multiplace regular representations are similar to those
of the regular multiactions. If i = 1 or i = k, the multiplace
representation is called a right or left regular representation. If k is
even, the representation with i = k�2 + 1 is called a middle regular
representation. The case k = 2 was considered in BOROWIEC
ET AL. [2006], DUPLIJ [2013a] for ternary groups.
90
15 Matrix representations of ternary groups
15 Matrix representations of ternary groups
Here we give several examples of matrix representations for
concrete ternary groups BOROWIEC ET AL. [2006], DUPLIJ [2013a].
Let G = Z3 3 {0, 1, 2} and the ternary multiplication be
[ghu] = g − h+ u. Then [ghu] = [uhg] and 0 = 0, 1 = 1, 2 = 2,
therefore (G, [ ]) is an idempotent medial ternary group. Thus
ΠL(g, h) = ΠR(h, g) and
ΠL(a, b) = ΠL(c, d)⇐⇒ (a− b) = (c− d)mod 3. (15.1)
The calculations give the left regular representation in the manifestmatrix form
ΠLreg (0, 0) = Π
Lreg (2, 2) = Π
Lreg (1, 1) = Π
Rreg (0, 0)
= ΠRreg (2, 2) = Π
Rreg (1, 1) =
1 0 0
0 1 0
0 0 1
= [1] ⊕ [1] ⊕ [1], (15.2)
91
15 Matrix representations of ternary groups
ΠLreg (2, 0) = Π
Lreg (1, 2) = Π
Lreg (0, 1) = Π
Rreg (2, 1) = Π
Rreg (1, 0) = Π
Rreg (0, 2) =
0 1 0
0 0 1
1 0 0
= [1] ⊕
−1
2−
√3
2√3
2−1
2
= [1] ⊕
[−1
2+1
2i√3
]⊕[−1
2−1
2i√3
], (15.3)
ΠLreg (2, 1) = Π
Lreg (1, 0) = Π
Lreg (0, 2) = Π
Rreg (2, 0) = Π
Rreg (1, 2) = Π
Rreg (0, 1) =
0 0 1
1 0 0
0 1 0
= [1] ⊕
−1
2
√3
2
−
√3
2−1
2
= [1] ⊕
[−1
2−1
2i√3
]⊕[−1
2+1
2i√3
]. (15.4)
92
15 Matrix representations of ternary groups
Consider next the middle representation construction. The middle
regular representation is defined by
ΠMreg (g1, g2) t =
n∑
i=1
ki [g1hig2] .
For regular representations we have
ΠMreg (g1, h1) ◦ΠRreg (g2, h2) = Π
Rreg (h2, h1) ◦Π
Mreg (g1, g2) , (15.5)
ΠMreg (g1, h1) ◦ΠLreg (g2, h2) = Π
Lreg (g1, g2) ◦Π
Mreg (h2, h1) . (15.6)
93
15 Matrix representations of ternary groups
For the middle regular representation matrices we obtain
ΠMreg (0, 0) = ΠMreg (1, 2) = Π
Mreg (2, 1) =
1 0 0
0 0 1
0 1 0
,
ΠMreg (0, 1) = ΠMreg (1, 0) = Π
Mreg (2, 2) =
0 1 0
1 0 0
0 0 1
,
ΠMreg (0, 2) = ΠMreg (2, 0) = Π
Mreg (1, 1) =
0 0 1
0 1 0
1 0 0
.
The above representation ΠMreg of 〈Z3, [ ]〉 is equivalent to the
94
15 Matrix representations of ternary groups
orthogonal direct sum of two irreducible representations
ΠMreg (0, 0) = ΠMreg (1, 2) = Π
Mreg (2, 1) = [1]⊕
−1 0
0 1
,
ΠMreg (0, 1) = ΠMreg (1, 0) = Π
Mreg (2, 2) = [1]⊕
1
2−
√3
2
−
√3
2−1
2
,
ΠMreg (0, 2) = ΠMreg (2, 0) = Π
Mreg (1, 1) = [1]⊕
1
2
√3
2√3
2−1
2
,
i.e. one-dimensional trivial [1] and two-dimensional irreducible.
Note, that in this example ΠM (g, g) = ΠM (g, g) 6= idV , but
ΠM (g, h) ◦ΠM (g, h) = idV , and so ΠM are of the second degree.
95
15 Matrix representations of ternary groups
Consider a more complicated example of left representations. Let
G = Z4 3 {0, 1, 2, 3} and the ternary multiplication be
[ghu] = (g + h+ u+ 1)mod 4. (15.7)
We have the multiplication table
[g, h, 0] =
1 2 3 0
2 3 0 1
3 0 1 2
0 1 2 3
[g, h, 1] =
2 3 0 1
3 0 1 2
0 1 2 3
1 2 3 0
[g, h, 2] =
3 0 1 2
0 1 2 3
1 2 3 0
2 3 0 1
[g, h, 3] =
0 1 2 3
1 2 3 0
2 3 0 1
3 0 1 2
96
15 Matrix representations of ternary groups
Then the skew elements are 0 = 3, 1 = 2, 2 = 1, 3 = 0, and
therefore (G, [ ]) is a (non-idempotent) commutative ternary group.
The left representation is defined by the expansion
ΠLreg (g1, g2) t =∑ni=1 ki [g1g2hi], which means that (see the
general formula (14.10))
ΠLreg (g, h) |u >= | [ghu] > .
Analogously, for right and middle representations
ΠRreg (g, h) |u >= | [ugh] >, ΠMreg (g, h) |u >= | [guh] > .
Therefore | [ghu] >= | [ugh] >= | [guh] > and
ΠLreg (g, h) = ΠRreg (g, h) |u >= Π
Mreg (g, h) |u >,
so ΠLreg (g, h) = ΠRreg (g, h) = Π
Mreg (g, h). Thus it is sufficient to
consider the left representation only.
97
15 Matrix representations of ternary groups
In this case the equivalence isΠL(a, b) = ΠL(c, d)⇐⇒ (a+ b) = (c+ d)mod 4, and we obtain thefollowing classes
ΠLreg (0, 0) = Π
Lreg (1, 3) = Π
Lreg (2, 2) = Π
Lreg (3, 1) =
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
= [1] ⊕ [−1] ⊕ [−i] ⊕ [i] ,
ΠLreg (0, 1) = Π
Lreg (1, 0) = Π
Lreg (2, 3) = Π
Lreg (3, 2) =
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
= [1] ⊕ [−1] ⊕ [−1] ⊕ [−1] ,
ΠLreg (0, 2) = Π
Lreg (1, 1) = Π
Lreg (2, 0) = Π
Lreg (3, 3) =
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
= [1] ⊕ [−1] ⊕ [i] ⊕ [−i] ,
ΠLreg (0, 3) = Π
Lreg (1, 2) = Π
Lreg (2, 1) = Π
Lreg (3, 0) =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
= [1] ⊕ [−1] ⊕ [1] ⊕ [1] .
Due to the fact that the ternary operation (15.7) is commutative,there are only one-dimensional irreducible left representations.
98
15 Matrix representations of ternary groups
Let us “algebralize” the regular representations DUPLIJ [2013a].From (7.15) we have, for the left representation
ΠLreg (i, j) ◦ΠLreg (k, l) = Π
Lreg (i, [jkl]) , (15.8)
where [jkl] = j − k + l, i, j, k, l ∈ Z3. Denote γLi = ΠLreg (0, i),
i ∈ Z3, then we obtain the algebra with the relations
γLi γLj = γ
Li+j . (15.9)
Conversely, any matrix representation of γiγj = γi+j leads to theleft representation by ΠL (i, j) = γj−i. In the case of the middleregular representation we introduce γMk+l = Π
Mreg (k, l), k, l ∈ Z3,
then we obtain
γMi γMj γ
Mk = γ
M[ijk], i, j, k ∈ Z3. (15.10)
In some sense (15.10) can be treated as a ternary analog of theClifford algebra. Now the middle representation is ΠM (k, l) = γk+l.
99
15 Matrix representations of ternary groups
Acknowledgements
I am deeply thankful to A. Borowiec, L. Carbone, D. Ellerman,
W. Dudek, A. Galmak, M. Gerstenhaber, S. Grigoryan,
V. Khodusov, G. Kurinnoy, M. Putz, T. Nordahl, B. Novikov,
Ya. Radyno, V. Retakh, B. Schein, J. Stasheff, E.Taft, R. Umble,
A. Vershik, G. Vorobiev, A. Voronov, W. Werner, and C. Zachos for
numerous fruitful discussions.
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