Leamer%20M.J.%20Groebner%20finite%20path%20algebras%20(M.S.%.pdf

8/14/2019 Leamer%20M.J.%20Groebner%20finite%20path%20algebras%20(M.S.%.pdf

1/33

Grobner Finite Path Algebras

Micah J. Leamer

Thesis submitted to the faculty of

Virginia Polytechnic Institute and State University

in partial fulllment of the requirements for the degree of

Master of Science

in

Mathematics

Edward Green, Chair

Charles Parry

John Rossi

July 1, 2004

Blacksburg, Virginia

Keywords: Groebner Bases, Path Algebra

Copyright 2004, Micah J. Leamer


2/33

Grobner Finite Path Algebras

Micah J. Leamer

Abstract

Let K be a eld and a nite directed multi-graph. In this paper I

classify all path algebras K and admissible orders with the property that

all of their nitely generated ideals have nite Grobner bases.


3/33

Acknowledgements

First of all I would like to thank Charles Parry and John Rossi for joining my

committee on such short notice. They exemplify the supportive environment that

the VA Tech math department fosters. Joseph Ball offered up his time and his

expertise to meet with me for several weeks and teach me the nuances of opera-

tor algebras. He was a faithful committee member until the rescheduling of my

defense conicted with his prior commitments. Adrian Keister offered me help

with many latex problems. Hannah Swiger has been invaluable. She has gone

above and beyond the line of duty to help me process forms and meet all of the

administrative requirements. She has been positive, reassuring and at all times

candid. I would especially like to thank Dean Reiss for encouraging me to join

the graduate school at Tech. He also opened doors so that my application processwas smooth and simple. I would not have spent this year at Tech nor written this

thesis otherwise.

Edward Green the chair of my committee has been a core gure in my math-

ematical development, as well as the central resource for this thesis. He has con-

sidered me valuable enough to meet with on a weekly basis for the past two years.

All of our meetings for the past year have been for the purpose of this thesis. I am

a better communicator and certainly much better at explaining my mathematical

ideas because of him. He has time and again listened to my theoretical babble and

helped me nd the gems therein. I have seen how he has balanced a successful

career in mathematics with an adept social intelligence. The marriage of these

two skills is rare. He introduced me to Grobner bases and path algebras. He has

guided me in my paper writing skills. He has reviewed multiple drafts of this

paper from my original chicken scratches to its current state. He has been both

capable at noting aws in proof and humble enough to correct my grammatical

mistakes. Most importantly he has been patient enough to put up with the eccen-

tricities of a young mathematician, who is full of himself.

There have been an endless number of other mathematicians who have aided

in my development, showed me kindness, and treated me as much like a colleague

as a pupil. Among the best of these are: Glen van Brummelen, Ezra Brown, GailLetzter, Neil Calkin and Peter Haskell.

iii


4/33

Contents

1 Introduction 1

2 The Fundamentals of Path Algebras 2

2.1 Examples of Path Algebras . . . . . . . . . . . . . . . . . . . . . 3

3 Path Orders 4

3.1 Examples of Admissible Orders: . . . . . . . . . . . . . . . . . . 5

4 Prerequisites for Grobner Theory 5

5 Basics of non-commutative Grobner theory 7

6 Induced Subgraphs 12

7 Classifying Noetherian Path Algebras 14

8 Producing Non-commutative Grobner Bases 16

9 Classifying Grobner Finite Path Algebras 18

iv


5/33

1 Introduction

The rst half of this paper may serve as a brief introduction to path algebras and

non-commutative Grobner bases theory. Nothing is assumed of the reader other

than a general understanding of algebra and graph theory, at the graduate level.

For now it will sufce to know that path algebras are a type of non-commutative

algebra over a eld. In particular, given a eld Kand a nite directed multi-graph, the path algebra K is the set of all K-linear combinations of paths of nitelength on. We give a concise denition of path algebras in section 2. Sections 3-5 introduce some key concepts in non-commutative Grobner theory, all of which

are well known results. Given an ordering


6/33

results of Grobner bases theory: Bergmans diamond lemma [2].

Given a nitely generated ideal in a path algebra it often occurs that any

Grobner basis of that ideal will be innite, regardless of what ordering is being

used. It is known that deciding whether an idealI K has a nite Gr obnerbasis is unsolvable in general. It is often necessary to rst nd a nite Grobner

basis to show one exists, which is not always true. Our main result, theorem 9.9, is

a classication of all path algebra, admissible order pairs which have the property

that all of their nitely generated ideals have nite Grobner bases. We dene a

path algebra with this property for some admissible order to be a Grobner nite

path algebra. The paper is structured such that the concepts in each section build

upon one another leading up to the climactic main result theorem 9.9 in section 9.

2 The Fundamentals of Path Algebras

Let = (0, 1)be a nite directed graph. We allow for to have arrows froma vertex to itself and multiple arrows between the same set of vertices. Through-

out the paper when referring to a graph we will assume it to be a nite directed

graphs of this type. We let 0 = {v1, v2, . . . , vN} be the set of vertices and1 = {A1, A2, . . . , AM}be the set of arrows. Arbitrary arrows in 1 will be de-noted byi, which need not equal Ai. Extending the same notation, letk be theset of paths of lengthk. Let=

i=0ibe the set of paths of, of nite length.

We dene functionso: 0andt: 0, such that for any pathp ,o(p)is the origin or rst vertex of the path p and t(p)is the terminus or nal vertex of

p. Note that, for any vertexv, o(v) = t(v) = v. For any pathp, we dene l(p)thelengthofp, to be the number of arrows that occur in p, counting multiplicities.We will say that two paths intersect if they share a common vertex. We denote

p= 12. . . r, whenpis a path of lengthl(p) =r >0. Wheneverpis a vertex,equivalently whenl(p) = 0, we denotep= vi. Our convention is such that a path12 n is written from left to right, more preciselyt(i) =o(i+1).

We dene multiplication of paths, such that for all p, q in , ift(p) = o(q),

thenpqis the path adjoiningp and q by concatenation. Otherwise, ift(p)=o(q)thenpq = 0. Given this denition, {0} is closed under multiplication. LetKbe an arbitrary eld. The path algebra, K, is dened to be the set of all nitelinear combinations of paths in with coefcients in K. Addition inK is theusualKvector space addition, where is aK-basis forK. Multiplication in

2


7/33

the path algebra, K, extends from the denition for multiplication of paths in

{0}. In general,Khas identity 1 =n

i=1vi andKn

i=1vi is containedin its center. ThusKacts centrally onK, so thatk

kipi=

(kki)pifor anyk, ki Kandpi . These operations makeKaK-algebra.

2.1 Examples of Path Algebras

Let be the graph with n vertices and no arrows. ThenK n

i=1K, withcomponentwise multiplication.

Let be the graph v1A1v2

A2. . . An1vn . ThenKis isomorphic to the

set ofn nupper triangular matrices overK. Numbering the vertices from 1 tonrespectively, we may dene an isomorphism sending a pathp to ann nmatrixwith a 1 in the(o(p), t(p))position and zeros in all other entries of the matrix.

We dene a loop to be an arrow from a vertex to itself. The path algebra K,where is the graph with one vertex and m loops is the free algebra in m non-commutative variables,K[x1, x2, . . . , xm].

In general,0 is a full set of primitive orthogonal idempotents for K. Twoidempotents are orthogonal if their product in either order is 0. A primitive idem-

potent is an idempotent that cannot be written as a sum of two orthogonal non-zeroidempotents.

Let Kl denote the set of all K-linear combinations of paths inof lengthl. Let a Ki and b Kj . Thenab is the product of a K-linear combina-tion of paths of length i times a K-linear combination of paths of length j. Itfollows that ab is a K-linear combination of paths of length i+ j . ThereforeKiKj Ki+j . ThusK

i=0Kl is a grading ofK.Givend N,

it follows that

d|lKl is a subalgebra ofK.

It is possible to represent every nite directed graph with nvertices as annn

matrix over the non-negative integers such that, the number of arrows from vertexi to vertexjis the(i, j)entry of the matrix. When there is any chance of confusionabout what graph is being discussed the matrix representation of the graph will be

given. Given a matrix representation for a graph, unless otherwise stated, we let

3


8/33

Aij(k)denote thekth arrow from vertexi to vertexj or, if the(i, j)entry is one,

we may simply writeAij .

3 Path Orders

We will now introduce certain orderings on, which will be relevant in develop-ing a Grobner basis theory for path algebras. A total ordering


9/33

3.1 Examples of Admissible Orders:

The left length-lexicographic order: Order the vertices and the arrows.

v1 < v2< . . . < vN < A1 < . . . < AM, such that the vertices are less than thearrows. Ifp and qare paths of length at least 1, then l(p) < l(q) impliesp < q.Otherwise, ifl(p) = l(q) = r thenp < q, wheneverp lex q. Note that wellordering does not fail since there are only a nite number of paths of any given

weight.

The total lexicographic order: Order the arrows arbitrarily A1 < . . . < Am.Also order the vertices. The vertices will be less than all paths of positive length.

Let p, q . Thenp < q, if there exists i such that j < i, Aj occurs in pandqthe same number of times, andAi occurs inp less than it occurs in q. Ifpandqhave the same number of each arrow then p


10/33

and for allp Supp(x),p T ip(x).

For reference, the equivalent of the tip in the commutative theory of Grobner

bases is called the head or leading term.

Denition 4.3 GivenX Kthe set{p |p = T ip(x)for somex X} isdenoted asT ip(X).

LetIbe an ideal in a path algebra K, with admissible ordering< on. Ifp T ip(I)and q thenqp, pq T ip(I)wheneverqp, pq= 0respectively.

Proposition 4.4 T ip(I)is aK-basis for the monomial ideal that it generates in

K.

Proof: Letp T ip(I), q , andqp = 0. Then there existsx I, such thatT ip(x) = p. Letx = p +

ni=1ipi withp, pi and, i K. Letq ,

such that t(q) = o(p). Thenqx = qp+n

i=1iqpi I. By the principles ofan admissible orderingp > pi fori = 1implies thatqp > qpi forqpi = 0. Thusqp = T ip(qx) T ip(I). Sop T ip(I), q impliesqp T ip(I). A similarargument shows thatpq T ip(I)for allq , such thatt(p) =o(q).

Corollary 4.5 Span(T ip(I))is an ideal ofK.

The set elements ofthat are not the tip of any element ofIis denotedNonTip(I) =T ip(I). Since=T ip(I) NonTip(I)is aK-basis forKand T ip(I)NonTip(I) =, it follows that Span(T ip(I)) and Span(NonTip(I))provide a direct sum decomposition ofKas aK-vector space.

Proposition 4.6 Let be a graph, K a eld and< an admissible order on .For each idealI inKK Span(T ip(I)) Span(NonTip(I))andK I Span(NonTip(I))as K-vector spaces.

Proof: Let x be a nonzero element in Span(NonTip(I))). Then T ip(x) NonTip(I). Thus x / I. Thus I

Span(T ip(I)) = . It follows that the

natural map fromX = I Span(NonTip(I))to K is injective. Assume thatthe natural map is not also surjective. By well ordering, we may let p be thesmallest path in that is not also in the image ofX. Thenp T ip(I). Thenthere exists z I such that, T ip(z) = p. Since p is the smallest path not inIm(X) it follows that Supp(z){p} Im(X). Thenz I Im(X) and

6


11/33

Supp(z){p} X, which implies p Im(X), a contradiction. Thus the nat-

ural map from X = ISpan(NonTip(I)) toK is surjective and the resultfollows.

It follows that every elementxofKmay be uniquely written, asix+N(x),withix IandN(x) Span(NonTip(I)). We callN(x)the normal formofx.

Denition 4.7 Forp, q , we sayp dividesq= 0, ifq= xpy for some pathsx, y .

5 Basics of non-commutative Grobner theory

We are now ready to introduce Grobner bases for path algebras. For a given ideal

Iin a path algebraK, the Grobner basisGofIis dependent upon the ordering


12/33

x0 I. Since< is a well ordering, every sequence of pathsp0 p1 p2

stabilizes. So the sequence of pathsT ip(x0) > Tip(x1) > must terminate,withxi = 0, for somei


13/33

Input:{f1, f2, . . . , f n}

Output:RS={f1, f2, . . . , f n}R= WHILE (S=)

choosefSS= S{f}S =R =WHILE (S=)

choosex SS=S{x}

S =S {Red{f}(x)}WHILE (R=)

choosex TR= R{x}IF (x= Red{f}(x))

R =R {x}ELSE IF (Red{f}(x)= 0)

S =S {Red{f}(x)}S= S

R= R {c1Tip(f)f}

We will denote the particular monic reduced generating set that this algorithm

produces from a generating setS, asR(S). The order< will be made apparentby the context.

Proposition 5.5 Givenx IandGa Grobner basis forI, every sequence of re-ductions ofxby Gterminates, withxtotally reducing to zero after a nite numberof reductions.

Proof: Let Ibe an ideal in a path algebra K, with admissible ordering < andlet G be a Grobner basis for I. The result of reducingx Iby an element of

Gis another element in I. Since the only element ofIthat cant be reduced byGis zero, we must show that every series of reductions of an element ofI byGterminates in a nite number of steps. Let


14/33

LetS= {s1, . . . , si} andT ={t1, . . . , tj} be nite sets of paths, ordered from

greatest to least. We sayS < T provided that there existsn N, n |T|, suchthatsk = tk, for allk < nand eithersn< tnor |S|= n 1. Let(Si)i=1be a non-increasing sequence of nite ordered sets of paths. Let(s ij)i=1 be the sequenceof thej th largest elements in each of the setsSi, withsij = 0, whenever |Si|< j.Then(si1)

i=1is a non-increasing sequence of paths and zeros and must, therefore,

stabilize to a patht1 or to zero. Provided that (sij)i=1stabilizes to a path tj , then(si(j+1))

i=1 must be non-increasing, after (sij)

i=1 stabilizes. Hence,(si(j+1))

i=1

must also stabilize to a pathtj+1< tj or to zero. The sequencet1, t2, . . .is strictlydecreasing so long as ti is not zero. Thus(ti)

i=1 must stabilize to 0. It follows

that the cardinality of the setsSiis uniformly bounded and that(Si)i=1must alsostabilize. Lety I.

Letx be a reduction ofy byg G. ThenSupp(x) < Supp(y). Hence, thesupports of the elements of a sequence of reductions, of an element y I, is astrictly decreasing sequence and must terminate after a nite number of reduc-

tions with the support of the total reduction being . Hence every sequence ofreductions must terminate at 0 in a nite number of steps and the result follows.

Proposition 5.6 Given an ideal I inK and an admissible order


15/33

Denition 5.7 Given an ideal I inK and an admissible order


16/33

dividesp. Thus,g may be reduced by g . Hence,g may be reduced by G{g},

which contradicts that Gis reduced and the result follows.

As in the commutative case, having a Grobner basis, for an ideal, allows one

to answer certain questions. It has been asked, given an ideal Iand an elementxinK is there a simple criteria to show whether or notx I? The solution isthatx I, if and only if, a full reduction ofx by a Grobner basis of the ideal is0. Furthermore, a Grobner basis allows us to represent the elements ofKI,uniquely, as the normal form of the elements, in the equivalence classes ofKmoduloI. In depth applications for Grobner bases may be found in [1], [3], [7]and [10].

6 Induced Subgraphs

Denition 6.1 Let be a graph andv i, vj vertices in. Then the induced sub-graph betweenvi andvj, denoted(vi, vj), is the smallest subgraph of, con-taining all the paths from vi tovj . If there are no paths betweenvi andvj, then(vi, vj)is the empty graph.

Denition 6.2 A maximal induced subgraph is an induced subgraph (v i, vj),such that(vi, vj) (vh, vk)implies(vi, vj) = (vh, vk).

We will see that it is possible to decompose a generating set or a Grobnerbases for an ideal on a path algebra over a graph into parts, which are restricted

to the induced subgraphs. What becomes useful is that we may rst construct

Grobner bases for an ideal restricted to an induced subgraph and then take their

union to produce a Grobner bases for the original ideal. Thus we may compart-

mentalize the problem of producing Grobner bases. This will not effect the speed

of computation but it will allow us to better classify Grobner bases in sections 7

and 9.

Denition 6.3 LetSbe a subset ofKand let be a subgraph of. Then therestriction ofSto , isS| ={x S| x K

}.

Notice that for idealsIKand subgraphs of,I| is an ideal inK.

An admissible order on a set of paths remains admissible on any of its sub-sets. Hence for any path algebraK, with admissible order


17/33

of then< remains an admissible order on K. For the remainder of the paper

we will assume that the order used on any subgraph is the same as the order usedon the original graph.

Proposition 6.4 Given a path algebraK, an admissible order< and an idealI inK, letGbe the reduced Gr obner basis forI. Let(vi, vj) be an inducedsubgraph of. Then G|(vi,vj ) is the reduced Gr obner basis forI|(vi,vj ).

Proof:Greduces the elements ofI|(vi,vj ) Ito zero. A uniform element x K

may reduce an element in K(vi, vj)only ifx K(vi, vj). G|(vi,vj ) reduces all

elements in I|(vi,vj ) to zero. ThusG|(vi,vj ) is a Grobner basis for I|(vi,vj) . If

g G|(vi,vj )

, theng Gand is fully reduced by G{g}. Thusg is fully reduced

by G|(vi,vj ){g} and is monic. So by proposition 5.6G|(vi,vj ) is the reduced

Grobner basis forI|(vi,vj ).

Corollary 6.5 LetIbe an ideal in the path algebra K and< an admissibleorder. Let be a subgraph of, which is the union of induced subgraphs of.

LetG be a reduced Gr obner basis for I and letG be a uniform Gr obner basesforI. Then G| is a reduced Gr obner basis forI| andG

|

is a uniform Gr obner

bases forI| .

Corollary 6.6 Given a path algebra K, an admissible order


18/33

Proposition 6.9 An ideal I in K is nitely generated, if and only if, I|(i)

is

nitely generated, for each maximal induced subgraph (i)

of.

Proof:SupposeI is a nitely generated ideal inK. LetSbe a nite generatingset for I. ThenUS =

xSUx is also nite, since |Ux| |0|

2. Let (i) be

a maximal induced subgraph, of. Then{x US|o(x), t(x) (i)0 } is a nite

generating set, for I|(i)

. Let (1), (2), . . . , (n) be the maximal induced subgraphsof. Suppose thatI|

(1), I|

(2), . . . , I |

(n)are nitely generated. LetS1, S2, . . . , S n

be nite generating sets, for each of the restricted ideals. Let fI. Then,y Ufimplies there existsi, such that y I|

(i) = Si. Sof S1, S2, . . . , S n and

consequentlyI S1, S2, . . . , S n. SinceSi Iwe haveI = S1, S2, . . . , S n.ThusIis nitely generated.

7 Classifying Noetherian Path Algebras

Denition 7.1 A graph is called Noetherian (not Noetherian) if its correspond-

ing path algebra is Noetherian (not Noetherian).

Proposition 7.2 A graph containing two nonidentical cycles that intersect at a

vertex is not Noetherian.

Proof: Let be a graph containing cyclesC andD that intersect at a vertex v i.Let c be a path of positive length alongC, from v i tovi, and let d be a path ofpositive length alongD, fromvito vi. Then, cdc, cd2c,cd3c , . . . is a non-nitelygenerated ideal ofK. SoKis not Noetherian.

Let = p1

p2

p3

p4

Let be a graph containing a graph of type as a subgraph, such thatp1, p3, p4 are all paths of positive length and p2 is possibly a path of length 0.Letqi = p1p2(p3p2)

ip4 thenq0, q1, q2, . . . is a non-nitely generated ideal, of

K. It follows that, any graph, which contains a cycle, with an arrow going intothe cycle and an arrow (possibly the same arrow) coming out of the cycle is not

Noetherian. Note that this includes the case with one vertex and two loops.

14


19/33

Proposition 7.3 A graphis not Noetherian, if and only if, it contains a cycleC,

an arrow not occurring inC, with its origin on C, and an arrow not occurring inC, with its terminus on C.

Proof: We have already shown that all graphs containing a cycle, with an arrow

going in the cycle and an arrow coming out of the cycle, are not Noetherian. It

remains to be shown that all other graphs are Noetherian. Let be a graph thatdoes not contain a cycle with an arrow entering the cycle and an arrow coming

out of the cycle. Then, all of the maximal induced subgraphs of contain atmost 2 cycles which do not intersect, one with arrows coming out of it and one

with arrows entering it. Let(i) be a maximal induced subgraph, of. Then(i), consists of two cyclesAand B, with no vertices in common andm possibly

overlapping paths fromA to B , allowing forA and B to be trivial cycles of onevertex.

(i) =

...

A

p2

p1

pm

B

Assume that, for somei,K(i) contains an idealI, which is not nitely gen-erated. LetSbe a reduced generating set for I. Then,Sis innite and the tips ofthe elements ofSdo not divide one another. Thus T ip(S)is also innite. LetTbe the nite set of paths that do not completely go around either of the cycles. For

eachp T letTp be the set of all paths with origin o(p) and terminus t(p) thatpdivides. Since every path on (i) is divisible by some path inT, with the sameorigin and terminus, it follows that there existsp Tsuch thatT ip(S)Tpis in-nite. Suppose, eithero(p)is not onAort(p)is not onB. Letq1, q2 T ip(S)Tp.Then the shorter ofq1 andq2 divides the other. SinceT ip(S) was reduced, thisimpliesq1 = q2and T ip(S) Tp has cardinality 1.

Thus we may assume, that o(p) is on A and t(p) is on B. For each pn T ip(S) Tp there exist non-negative integersan andbn such thatpn wraps com-pletely aroundA, an times andpn wraps completely aroundB, bntimes. Thenpndoes not dividepm implies either an > am orbn > bm. Since there are onlyan

non-negative integers less thanan and bn non-negative integer places less than bnthen |T ip(S) Tp| an+ bn+ 1


20/33

8 Producing Non-commutative Grobner Bases

In the commutative case, the Buchberger algorithm, for computing a Grobner

basis, relies upon computing S-polynomials from pairs of polynomials. The non-

commutative version of the S-polynomial is the overlap relation. The algorithm

that we introduce in this section will similarly rely upon overlap relations to pro-

duce a Grobner basis. Letf, g K, with admissible order < on K. Supposethere are pathspandq, of positive length, such that T ip(f)p= qTip(g), with thelength ofpless than the length ofT ip(g). Thenf andg have anoverlap relation,denotedo(f,g,p,q ), given by

o(f,g,p,q ) =c1Tip(f)f p c1Tip(g)qg .

In the non-commutative case, given polynomials f andg that overlap, thep andqwill not necessarily be unique and consequently neither will the overlap relation.

Example: LetK = K[x, y] be the free algebra in two non-commutative vari-ables. Let < be the length lexicographic order, with x < y . Let f= 5yyxyx2xxandg = xyxy 7y. ThenT ip(f) =yyxyxand T ip(g) =xyxy. There are threeoverlap relations.

o(f,g,y,yy) = (1/5)f y yyg= (2/5)xxy+ 7yyy

o(f,g,yxy,yyxy) = (1/5)fyxy yyxyg= (2/5)xxyxy+ 7yyxyy

o(g,g,xy,xy) =gxy xyg = 7yxy + 7xyy

Bergmans Diamond Lemma [2] 8.1 LetG be a set of uniform elements thatform a basis for the idealI K, such that for allg, g G,Tip(g) does notdivideT ip(g). Suppose that for eachf, g Gevery overlap relationo(f,g,p,q )reduces to 0 by G. Then, Gis a Grobner basis for the idealI.

Proof: Gis a spanning set. So for each nonzerox I, we may represent x asx =

i,jcijpjgiqj where gi G, pj , qj , and cij K. Since multiples of

elements ofGsum to zero, this representation is in no way unique. Given such arepresentation, for an elementx, letp be the largest path in the support of any of

the elementspjgiqj . Let us choose a representation ofx, so thatpis the small-est possible. Among the representations wherepis the smallest possible, let uschoose a representation wherep occurs in the least number of terms possible. Itfollows thatp= T ip(pjgiqj)for somei andj, in the representation.

16


21/33

Assume there are n > 1 pairs i, j such that p = T ip(pjgiqj). Let i, j and

i

, j

be two such pairs. For notational convenience let us writep = pj , g = gi,q = qi, p = pj , g

= gi andq = qi . Thenp = T ip(pgq) = pTip(g)qand

p = T ip(pgq) = pT ip(g)q. Recall thatl(s) is the length of any path s. Ifl(p) = l(p), then one ofT ip(g) or T ip(g)would have to divide the other, con-tradicting the hypothesis for G. So without loss of generality, we may assume thatl(p)> l(p).

Ifl(p) l(pT ip(g)), then there exists s , such thatp = pT ip(g)s andq =sTip(g)q. Letg =kT ip(g) +

kiaiand letg

=Tip(g) +

ibi, suchthata, ai, b , bi andk, ki, , i K.

pgq = p

T ip(g

)sgq= p(T ip(g) +

ibi)sgq

ip

bisgq

= 1pgsgq

ip

bisgq

= 1pgs(kTip(g) +

kiai)q

ipbisgq

= kpgsTip(g)q+

kipgsaiq

ipbisgq

= kpgq +

kip

gsaiq i

pbisgq

p occurs in kpgq and in none of the other terms. All of the other terms only

have smaller paths occurring in them. So it follows that we may representx sothat the largest path p occurs in less than n terms in the representation. This

contradicts the hypothesis that our representation has p occurring in a minimalnumber of terms. So the assumption that poccurs in more than one term of therepresentation is false. It follows thatp= T ip(x)andxmay be reduced.

Ifl(p) < l(p) < l(pT ip(g)). ThenT ip(g) and T ip(g) overlap in p. Sothere exists an overlap relationo(g, g , r , s) =c1Tip(g)g

r c1Tip(g)sgwithr, s ,

such thatp= psandrq= q.

pgq = psgq

= cTip(g)c1Tip(g)p

grq

cTip(g)c1Tip(g)p

grqpsgq

= cTip(g)c1

Tip(g)p

g

rq cTip(g)p

c1

Tip(g)g

r c1

Tip(g)sg

q= cTip(g)c

1Tip(g)p

gq cTip(g)po(g, g , r , s)q

Only paths less thanpoccur inpo(g, g , r , s)qando(g, g , r , s)may be reducedto zero by G. So it follows that we may represent x so that the largest path p

17


22/33

occurs in less thann terms in the representation. This contradicts the hypothesis

that our representation has p occurring in a minimal number of terms. So theassumption thatp occurs in more than one term, of the representation is false. Itfollows that,p = tip(x)and x may be reduced. Thus,G reduces every elementofIto zero and Gis a Grobner basis.

The Buchberger-Mora-Farkas-Green Algorithm [5] 8.2 Given a path algebra

Kan admissible order< and a nite generating set{f1, f2, . . . , f m}for an idealIthe following algorithm gives a reduced Gr obner basis forIin the limit.

Input: A generating set {f1, f2, . . . , f m} for an idealIK

Output: A reduced Grobner basis in the limit Gn = {g1, g2, . . .}n= 0G0 = G1 = R({f1, . . . , f n})WHILE (Gn=Gn+1)

n++

Gn= GnFOR ( all pairsf, g Gnand all overlap relationso(f,g,u,v))

Gn= Gn {o(f,g,u,v)}

Gn+1= R(Gn)

The property that this algorithm produces a reduced Grobner bases in the limit

will be useful in the next couple of proofs. In particular, we will use the fact that

the algorithms only performs overlap relations and reduction in order to produce a

Grobner basis. To produce a Grobner basis in the limit means that, if the algorithm

stops on thenth iteration the set Gnis a reduced Grobner basis. Otherwise, given areduced Grobner bases G, for allx Gthere existsk N, such that for alln kwe have{g Gn|g x} = {g G| g x}. For a proof that the algorithmproduces a Grobner basis in the limit, the interested reader may peruse [5].

9 Classifying Grobner Finite Path AlgebrasDenition 9.1 We will say that a path algebra is Grobner nite if there is an

admissible order


23/33

Our next goal will be to classify all Gr obner nite path algebras. Furthermore,

we will show that if a path algebra is not Grobner nite then it contains nitelygenerated ideals whose reduced Grobner basis is not nite under any admissible

order. We show this in our main result theorem 9.9. All of the materials in this

section are either used directly in the proof of theorem 9.9 or they are implied by

the main result. Propositions 9.6 and 9.7 are direct results of theorem 9.9 and may

be omitted.

Example 9.2 Letbe the graph given below.

p

b

b

q

c

Let< be an admissible ordering on the paths of such thatqci > bbq, then theideal pbq, qci bbq Khas innite reduced Gr obner basis {pb(bb)jq,qci bbq|j N}.

Example 9.3 Letbe the graph given below.

p1

p2

p3

p4

p5

p6

p7

Let< be an admissible ordering on the paths of. Ifp4p5p6 > p3p2p4, thenthe ideal p1p2p4, p4p5p6 p3p2p4 K has innite reduced Gr obner ba-sis {p1p2(p3p2)ip4, p4p5p6 p3p2p4| i N}. Else if p3p2p4 > p4p5p6, thenthe ideal p4p5p7, p3p2p4 p4p5p6 K has innite reduced Gr obner basis{p4(p5p6)ip5p7, p3p2p4p4p5p6| i N}. It follows thatKhas a nitely gener-ated ideal with an innite reduced Gr obner basis, under any admissible ordering.

Example 9.4 Now let be a graph which contains two cycles P andQ, whichintersect at a vertexv. Letpbe the path fromv to itself along cyclePand letqbethe path fromv to itself along cycle Q. Ifpq2 > p2q, then the idealpqpq, pq2

p2

q K has Gr obner basis {pqpi

q,pq2

p2

q| i N}. Else ifp2

q > pq2

,then the ideal pqpq, p2qpq2 K has Gr obner basis {pqipq, p2qpq2|i N}. Therefore, any path algebra which contains two intersecting cycles, alsocontains a nitely generated ideal with and innite reduced Grobner basis under

any admissible ordering.

19


24/33

Denition 9.5 LetA be an arrow in the path algebra K. Then, for any pathp,

degA(p), the degree ofA inp is the number of timesA occurs inp. Furthermore,for anyx K,degA(x) =max{degA(p)|p Supp(x)}

Proposition 9.6 Letbe the graph

v1a v2

b

c v3

d

with matrix representation

0 1 00 1 10 0 1

Let< be an admissible order. Then there exists a nitely generated ideal ofKwith innite reduced Gr obner basis, if and only if, there exists j > 0 such that

cdj > bc.

Proof:If there existsj >0, such thatcdj > bc, then the ideal ac,cdjbc Khas innite reduced Grobner basis {abic,cdj bc|i N}. Instead, suppose for all

j >0 we havecdj < bc. Assume there exists a nitely generated idealIK,such that G, the reduced Grobner basis for Iis innite. By proposition 5.10, the el-ements ofGare uniform. Letp, qbe paths on such thatdegb(p) =degb(q)= 0.Suppose thata occurs in neither p nor q. Then the longer of the two divides theother. Similarly ifa occurs in bothp and qthen the longer of the two divides theother. Therefore, if neitherp nor qdivides the other, thena must occur in exactlyone ofporq. Thus every reduced set of paths, all of which have the same nonzerodegree ofb, has at most 2 elements. Similarly a reduced set of paths all with de-greebequal to zero has at most 3 elements , {a,cdi, dj}, withj > i. Consequentlygiven a reduced set of pathsS,|S| 2suppS{degbp} + 3 (Heresup representsthe supremum not the support). It follows that the degree ofb for T ip(G) mustbe unbounded in order for Gto be innite. Consequently, the degree ofb in theelements ofGmust be unbounded.

Let x be a uniform element in K. Suppose o(x) = t(x) = v2. Thenx=

ni=1kib

i, for someki K, withkn = 0and degb(x) =degb(T ip(x)) =n.Suppose o(x) = v1 and t(x) = v2. Then x =

ni=1kiab

i and degb(x) =

degb(T ip(x)) = n. Suppose o(x) =v2and t(x) =v3and degb(x) =n, degd(x) =m. Let x =

ni=0

mi=0kijb

icdj , with kij K. Let h be maximal such thatknh = 0. Assume bncdh = T ip(x) = bicdj . Well ordering impliesbncdh doesnot divide T ip(x) = bicdj . This implies j > h. Since h was maximal i < n.Thus by well orderingbnic < cdjh. Furthermorebc < cdjh, which contradicts

20


25/33

our hypothesis. Thusbncdh = T ip(x) anddegb(T ip(x)) = degb(x) = n, Sup-

poseo(x) = v1 andt(x) = v3. Then there exists a uniform element x

, such thato(x) = v2 andt(x) = v3 andx = ax

. Thendegb(T ip(x)) = degb(T ip(x)) =

degb(x) = degb(x). In all other casesdegb(x) = degb(T ip(x)) = 0. Thus, forallx K,degb(x) = degb(T ip(x)). It follows that reducing an element cantincrease thedegbof that element. If there exists g Gsuch thato(g) =t(g) =v2,thendegb(G) = degb(g). So there exists no such element. If there existsg Gsuch that o(g) = v1 and t(g) = v2, then degb(v1G) = degb(g) and Gv1G iscontained within a Noetherian subgraph and is therefore nite. Thus Gis nite, acontradiction.

So we may assume that Gv2 = . Thus, for uniformx I either x = v1

or t(x) = v3. Let Gn be the set generated by the nth iteration of algorithm8.2 starting with a nite generating set for I. Then and Gnv2 = . In orderfor degb(G) to be unbounded there must exist k andg Gk+1 such that for allg Gk, we have degb(g) > degb(g). Since reductions do not increase thedegbof an element it must be that g is the full reduction of an overlap relation andnot just simply the reduction of an old element ofGk. Thusg is the full reduc-tion ofo(f,g,u,v) with f, g Gk. fhas an overlap relation implies f = v1.Thus t(f) = v3 and degb(v) = 0. T ip(f v +o(f,g,u,v)) = T ip(f v). Thusdegb(g

) degb(o(f,g,u,v)) degb(f v+o(f,g,u,v)) =degb(f v) =degb(f).This contradicts thatdegb(G)is unbounded and the result follows.

Proposition 9.7 LetK be a path algebra such that every path on intersectsat most 1 cycle. Then, given any admissible order, every nitely generated ideal

IKhas a nite Grobner basis.

Proof: Let be a graph withn cyclesCi, such that every path on intersects atmost 1 cycle, and let< be an admissible order. Note that there are graphs of thistype that are not Noetherian. Let Ibe a nitely generated ideal in K. LetGbethe reduced Grobner basis for I. LetG0 be a nite uniform generating set forI.Assume Gis innite. For alln N the set of paths inwithdegCi less thannforalli is nite. So we may assume there exists i such thatdegCi of the elements of

Gis unbounded.

Note that performing overlap relations between uniform elements and reduc-

ing uniform elements by uniform elements, produces uniform elements. Conse-

quently, at any stage in the algorithm, Gn is a uniform set. Let Sibe the nite set

21


26/33

of paths that either begin or end on Ci but do not traverse Ci at all. Lets Si

witho(s) Ci and let p = as andq = bs witha andb on Ci. Without loss ofgenerality we may assumea < band consequentlya dividesb and p dividesq. Itfollows that every reduced set of paths that all begin or end on Ci is nite. ThusviGvj is nite wheneverv ior vj is on one of the cycles.

Consequently, there existsvi andvj not on any cycle, such that viGvj is in-nite. We say a vertex vkis between viand vjif there is a pathp, such that o(p) =viandt(p) = vj withvk a vertex onp,vk =viandvk =vj . There must exist at leastone pairvi, vj not on any cycle withviGvj innite, such that, for allv k betweenviandvj , bothviGvk andvkGvi are nite. Starting with a nite generating set forIwe may reach a point in algorithm 8.2 such that viGvk Gn andvkGvi Gn

for allvk betweenvi andvj , then there are only a nite number of element inGnwhose overlap relations are inviKvj . After these overlap relations have beenreduced no other elements in viKvj will be produced by algorithm 8.2. ThusviGvj must be nite. This contradicts the assumption that Gwas innite.

Proposition 9.8 Suppose is a graph with no intersecting cycles and does notcontain a graph of the form

= p1

p2

p3

p4

p5

p6

p7

as a subgraph, such thatp1, p2, p4, p5, p7 are paths of positive length andp3, p6are paths of possibly length zero. Then a nonempty uniform subgraph(vi, vj),withvi=vj consists of a cycleA,ni N pathspij fromA to a cycleBi,mi N

pathsqij from cycleBi to a cycleC, fori, t N such thatj < t ands N {0}pathshi fromA to C. It is possible for either cycleA orCto be trivial cycles ofone vertex. Thepij may intersect and overlap one another, the qij may intersectand overlap one another and thehimay intersect one another and as a group theymay rst intersect thepij and then theqij . These restrictions on the intersectionsof thepij , qij andhi entail that no path on(vi, vj)may overlap more than one oftheBi.

Proof: Supposing a graph doesnt contain any intersecting cycles then everypathpon must be contained within a subgraph

C1

q1

C2

q2 qn1

Cn

22


27/33

Since does not contain the graph

= p1

p2

p3

p4 p5

p6

p7

pcan travel on at most 2 of the paths qi, it follows that eitherpis contained withinthe subgraphp, such that either

p=

A

pij

Bi

qik

C

or

p =

A

hi

C

Where A or Cmay trivial cycles consisting of one vertex. Additionallyo(p) ison A and t(p) is on C. If we take all the paths p from vi to vj that dont gocompletely around any cycle more than once, then (vi, vj) =

pvivj

p. Thecycles Bi and Bj do not intersect for i = j since contained no intersectingcycles. Furthermore, the paths pij may overlap and the pathsqij may overlap. Ifthepij , Bi orqij shared any vertices with another not of their type then the graph would have to contain a graph of the form , contradicting the hypothesis.Furthermore, if any of the pathshi intersect the pathspij orqij out of order, thenthe graphwould have to contain a graph of the form as a subgraph. The result

follows.

Theorem 9.9 A path algebraKwith admissible ordering< contains a nitelygenerated ideal with innite reduced Grobner basis if and only if the graphandthe order


28/33

wherel(b)may be zero, with pbb < aipfor somei.

Proof: Let K be a path algebra and < an admissible order. We have alreadyshown in examples 9.2 and 9.4 that if and < satisfy any of (1), (2), or (3) thenKcontains a nitely generated ideal with an innite reduced Gr obner basis. Sowe may assume that none of conditions (1), (2), or (3) are satised. Example 9.3

shows that if were to have a graph of the form

= p1

p2

p3

p4

p5

p6

p7

as a subgraph then it would have to satisfy either (2) or (3). So does not haveany graph of the form as a subgraph. LetIbe a nitely generated ideal in Kwith reduced Grobner basisG. Assume thatG is innite. Then, by proposition6.4, there exists an induced subgraph(vi, vj)of, such that G|(vi,vj ) the reduced

Grobner basis ofI|(vi,vj ) is innite.

By proposition 9.8 we know that(vi, vj) consists of a cycle A,ni pathspijfromA to a cycleBi,mi pathsqik from cycleBi to a cycleC, for alli N suchthati < t N ands N {0} pathshifromAtoC. It is also possible for eithercycleA or C to be trivial cycles of one vertex. The pathspij may intersect andoverlap one another, the paths qij may intersect and overlap one another and the

pathshi may intersect one another and they may as a group rst intersect thep ijand then theqij . Letaij be the path witho(aij) =t(aij) =o(pij)that goes aroundA once and letcik be the path with o(cik) = t(cik) = t(qik) that goes around Conce. In the case thatA orCis a trivial cycleaij = o(pij) and cik = o(pik) re-spectively. Also letbijk be the shortest path alongBifromt(pij)to o(qik)and letbijk be the path alongBi such thatbijkb

ijk completes one cycle aroundBi. Since

(2) and (3) are not satised we have p ijbijkbijk > ahijpij andb

ijkbijkqik > qikc

hik

for alli, j, kand h.

We may choose (vi, vj) to be minimal in the sense that for all (vh, vk)

strictly contained in (vi, vj), we have G|(vi,vj) is innite but G|(vh,vk) is nite.ThusvhGvkis nite whenevervhand vk are not both onAandCrespectively and(

viAvi)G(

vjC

vj)is innite. Let be the subgraph of(vi, vj)containingcyclesA, Cand only the pathshifromAtoCthat do not intersect any of theBi.Since is Noetherian every reduced set of paths on is nite. Therefore the

24


29/33

subset of(

viAvi)T ip(G)(

vjC

vj)containing only those paths which do not

intersect some cycleBiis nite.

We may partition the paths starting on A and ending onCthat intersect somecycleBi into sets,Spq = {ap(bb)

hbqc} = {p(bb)hb1qc| h N {0},a and care any path onA and Crespectively witht(a) = o(p)and o(c) = t(q)} wherefor somei, j, k we havep = pij, q= qik, b = bijk , and b = bijk . There are onlynitely many such sets. Therefore, there exists Spq, such that T ip(G) Spq isinnite.

Let R be a relation on the sets Spq, such that SpqRSpq whenever for allx Spq there existy Spq , such thatx < y. Suppose we dont haveSpqRSpq

then existsx Spq such that x > y for ally Spq , which implies SpqRSpq.It follows that for all pairs of sets Spq and Spq at least one ofSpqRSpq andSpqRSpq is true. Suppose thatSpqRSpq andSpqRSpq . Then for allx Spqthere existy Spq , such thatx < yand there existszSpq withx < y < z.ThusSpqRSpq . These properties of the relationR allow us to re-label the setsSpq as the sets Phk with h, k N, in such a way that PhkRPhk if and only ifh h. LetTh =

kPhk. Then there existsh such thatT ip(G) Th is innite.

Let be largest integer, such thatT ip(G) T is innite.

Let H be the nite subset of (

viAvi)G(

vjC

vj) consisting of thoseelements whose tips dont intersect any of the cycles

Bj. Starting with a nite

generating set let Gn be the set dened in the nth iteration of algorithm 8.2.We may reach a the point in algorithm 8.2, such that for all vh and vk notboth on A and C we have vhGvk Gn, H Gn, and {g G| T ip(g) Thfor h > } Gn. Furthermore let us choosen large enough so that all theoverlap relations o(f,g,u,v) with f, g H (

vi /A or vk /C

viGvk) {g G|T ip(g) Th for h > } reduce to zero by Gn.

It follows that if an overlap relation o(f,g,u,v) (

viAvi)K(

vjC

vj)

does not reduce to0, withf, g Gn, then eitherT ip(g) Th or T ip(f) Th forsomeh . Letx be the largest path in Gn (hTh). Since for allh we have PhjRP1there exists xn P1such that xn> x. Let xn= anp(bb)bqcn.

For f, g Gn let y = 0 be the element that o(f,g,u,v) (

viAvi)K(

vjC

vj) reduces to when the set Gn, of elements from Gn

25


30/33

union their overlaps, is made into the reduced set Gn+1 at the end of the nth

iteration of algorithm 8.2. Specically, we are referring to the line in algorithm8.2, Gn+1= R(Gn), where algorithm 8.2 calls algorithm 5.4. In general for a gen-erating setSevery element in the reduced setR(S)is the reduction of an elementinS. IfT ip(g) Phj for somej andh thenT ip(y) T ip(o(f,g,u,v)) p(bb)

+2bq. Any reduced setof paths in the set {ajpj(bjbj)

tbjqjcj |pj(bj1bj2)tbj1qj is a xed path}is nite. It

follows that any reduced set of paths in Th all of which are less than p(bb)+2bq

is a nite union of reduced set of paths from the sets {ajpj(bjb

j)t

bjqjcj|pj(bjbj2)tbj1qj is a xed path}. It follows that T ip(G) T is nite. Thiscontradicts the hypothesis that(

viA

vi)G(

vjCvj) is innite. ThusG|(vi,vj )

is nite, and by corollary 6.6, Gis nite.

We will now show an example of an admissible order on the path algebras

dened in proposition 9.8, which meets the criteria of the ordering described in

the theorem 9.9. Since such an order exists this implies the path algebras dened

in proposition 9.8 are all Grobner nite.

The left dual-weighted-lexicographic order: Let be a graph of the form

described in proposition 9.8. Let1 be the set of arrows which are containedwithin cycles in the graph which do not have both an arrow entering themand an arrow coming out of them. Fix a set of positive integers{n| N}Let W : 1 N N, such that for 1 we have W() = ( 0, n) and

26


31/33

for 1 we have W() = (n, 0). Dene W : N such that

W(1. . . r) =r

i=1W(i), with componentwise addition. Order the set NNso that(n, m) < (n, m) whenevern < n or whenn = n andm < m. Next,order the vertices and let W(vi) = 0 for all vi 0. Order the arrows so thati < j wheneverW(i)< W(j). Finally denep < q, ifW(p)< W(q), elseifW(p) =W(q), then use the left lexicographic order.

In our example above, the set N N is given a well order and there areonly a nite number of paths with given weight (n, n ). Therefore the leftdual-weighted-lexicographic order must also be a well order. We leave it to the

reader to show that the left dual-weighted-lexicographic order meets the criteria

of the order described in theorem 9.9 and that it satises the two conditions in

section 3 which make it an admissible order.

References

[1] Bardzell, Micheal J. (2001). Non-commutative Grobner bases and

Hochschild cohomology,Contemp. math.,286, 227-240.

[2] Bergman, G. (1978). The diamond lemma for ring theory. Adv. Math., 29,

178-218.

[3] Cojocaru Svetlana, Podoplelov Alexander, Ufnarovski Victor. Non-

commutative Grobner bases and Anicks resolution.Prog. math.,173, Verlag

Basel/Switzerland , 139-159.

[4] Cox, D., Little, J., OShea, D. (1992). Ideals, varieties, and Algorithms,

UTM Series. Springer-Verlag.

[5] Farkas, D., Feustal, C., and Green, E. L. (1993). Synergy In The theories of

Grobner bases and path algebras.Can. J. Math.,45, 727-739.

[6] Green, E. L. (1999). Non-commutative Grobner basis and projective resolu-

tions.Progress in mathematics,173, Verlag Basel/Switzerland, 29-60.

[7] Kronewitter F. Dell (2001). Using non-commutative Grobner bases in solv-

ing partially prescribed matrix inverse completion problems.Linear algebra

appl.,338, 171-199.

27


32/33

[8] Mora, Teo (1989). Grobner bases and non-commutative algebras. Lecture

notes in comput. sci.,358, Spring Berlin, 150-161.

[9] Mora, Teo (1994). An introduction to commutative and non-commutative

Grobner bases.Theoret. comput. Sci.,134, 131-173.

[10] Peeva Irena, Reiner Victor, Sturmfels Bernd (1998). How to shell a monoid.

Math. ann.,310, 379-393.

28


33/33

Index

Leamer%20M.J.%20Groebner%20finite%20path%20algebras%20(M.S.%.pdf

Documents