ThesisFinal a Graph Theoretic Approach to Combinatorial

A graph theoretic approach to combinatorialproblems in semigroup theory

Robert Gray

School of Mathematics and StatisticsUniversity of St Andrews

Thesis submitted for the Degree of Doctor of PhilosophyUniversity of St Andrews

January 27, 2006

Abstract

The use of graph theory has become widespread in the algebraic theory of semi-groups. In this context, the graph is mainly used as a visual aid to make presenta-tion clearer and the problems more manageable. Central to such approaches is theCayley graph of a semigroup. There are also many variations on the idea of theCayley graph, usually special kinds of subgraph or factor graph, that have becomeimportant in their own right. Examples include Schutzenberger graphs, Schreiercoset graphs and Van Kampen diagrams (for groups), Munn trees, Adian graphs,Squier complexes, semigroup diagrams, and graphs of completely 0-simple semi-groups. Also, the representation of elements in finite transformation semigroupsas digraphs has proved a useful tool.

This thesis consists of several problems in the theory of semigroups with the com-mon feature that they are all best attacked using graph theory. The thesis has twoparts. In the first part combinatorial questions for finite semigroups and monoidsare considered. In particular, we look at the problem of finding minimal generat-ing sets for various endomorphism monoids and their ideals. This is achieved bydetailed analysis of the generating sets of completely 0-simple semigroups. Thisinvestigation is carried out using a bipartite graph representation.

The second part of the thesis is about infinite semigroup theory, and in particularsome problems in the theory of semigroup presentations. In particular we con-sider the general problem of finding presentations for subsemigroups of finitelypresented semigroups. Sufficient conditions are introduced that force such a sub-semigroup to be finitely presented. These conditions are given in terms of theposition of the subsemigroup in the parent semigroups left and right Cayleygraphs.

i

Declarations

I, Robert Gray, hereby certify that this thesis, which is approximately 60,000words in length, has been written by me, that it is the record of work carriedout by me and that it has not been submitted in any previous application for ahigher degree.

Signed .................................... Date .................

I was admitted as a research student in September 2002 and as a candidate forthe degree of PhD in September 2003; the higher study for which this is a recordwas carried out in the University of St Andrews between 2002 and 2005.

Signed .................................... Date .................

In submitting this thesis to the University of St Andrews I understand that Iam giving permission for it to be made available for use in accordance with theregulations of the University Library for the time being in force, subject to anycopyright vested in the work not being affected thereby. I also understand thatthe title and abstract will be published, and that a copy of the work may be madeand supplied to any bona fide library or research worker.

Signed .................................... Date .................

I hereby certify that the candidate has fulfilled the conditions of the Resolutionand Regulations appropriate for the degree of PhD in the University of St An-drews and that the candidate is qualified to submit this thesis in application forthat degree.

Signature of Supervisor .................................... Date .................

iii

Acknowledgments

I would like to take this chance to thank those people who have made this thesispossible.

Firstly I would like to thank my family for the constant support and encour-agement they have provided over the years. I would especially like to thank mymother, father and my two sisters Heather and Mary.

Next I would like to thank my supervisor Professor Nik Ruskuc for his encour-agement and guidance during my PhD. I would also like to thank the rest of thestaff at the university for all the help that they have provided.

Finally I would like to thank all my friends and colleagues who have made thepast three years so enjoyable. I would especially like to thank Catarina, for proofreading my papers and thesis, Lizzie, for being a wonderful office mate, and Peter,for the interesting maths conversations at the breakfast table.

v

Contents

1 Preliminaries 11.1 Semigroup theory preliminaries . . . . . . . . . . . . . . . . . . . . 21.2 Graph theory preliminaries . . . . . . . . . . . . . . . . . . . . . . 9

I Finite semigroup theory 13

2 Minimal generating sets using bipartite graphs 152.1 Finite semigroups and their generating sets . . . . . . . . . . . . . 162.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Rectangular 0-bands . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Finite 0-simple semigroups and their associated graphs . . . . . . . 252.5 Connected completely 0-simple semigroups . . . . . . . . . . . . . 332.6 Arbitrary completely 0-simple semigroups . . . . . . . . . . . . . . 382.7 Substitution lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.8 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.9 Expressing rmin as a property of G . . . . . . . . . . . . . . . . . . 492.10 Isomorphism theorem and normal forms . . . . . . . . . . . . . . . 532.11 Applications, examples and remarks . . . . . . . . . . . . . . . . . 562.12 Non-regular Rees matrix semigroups . . . . . . . . . . . . . . . . . 602.13 A family of transformation semigroups . . . . . . . . . . . . . . . . 62

3 Idempotent generating sets and Halls marriage theorem 693.1 Semigroups generated by idempotents . . . . . . . . . . . . . . . . 703.2 Division and direct products . . . . . . . . . . . . . . . . . . . . . . 723.3 Idempotent generated completely 0-simple semigroups . . . . . . . 763.4 Regular and symmetric bipartite graphs . . . . . . . . . . . . . . . 863.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.6 Nilpotent rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.7 Counting generating sets . . . . . . . . . . . . . . . . . . . . . . . . 102

4 Free G-sets and trivial independence algebras 1114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2 Universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.3 Independence algebras . . . . . . . . . . . . . . . . . . . . . . . . . 116

vii

4.4 Generating sets for End(A) . . . . . . . . . . . . . . . . . . . . . . 1194.5 Generating sets for ideals of End(A) . . . . . . . . . . . . . . . . . 1234.6 General strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.7 Trivial independence algebras without constants . . . . . . . . . . 1264.8 Trivial independence algebras with constants . . . . . . . . . . . . 134

5 Vector spaces and non-trivial independence algebras 1435.1 Non-trivial independence algebras . . . . . . . . . . . . . . . . . . 1445.2 Non-trivial independence algebras with constants . . . . . . . . . . 1445.3 Non-trivial independence algebras without constants . . . . . . . . 1545.4 Proving the result directly from the definition . . . . . . . . . . . . 166

6 Large completely simple subsemigroups and graph colouring 1676.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.2 Left and right zero semigroups . . . . . . . . . . . . . . . . . . . . 1696.3 Completely simple semigroups . . . . . . . . . . . . . . . . . . . . . 173

II Infinite semigroup theory 183

7 Generators and relations via boundaries in Cayley graphs 1857.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1867.2 Examples and basic properties . . . . . . . . . . . . . . . . . . . . 1897.3 Generating subsemigroups using boundaries . . . . . . . . . . . . . 1947.4 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1967.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.6 One sided boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 2167.7 Left and right independence . . . . . . . . . . . . . . . . . . . . . . 2187.8 The converse: unions of semigroups . . . . . . . . . . . . . . . . . . 2207.9 Subsemigroups of free semigroups . . . . . . . . . . . . . . . . . . . 223

8 Strict boundaries and unitary subsemigroups 2298.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.3 Generating sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2328.4 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

viii

List of Figures

1.1 A graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 Two egg-box pictures . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Diagram showing that the groups Vjpj and Vipi are conjugate

in G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 Connecting coordinates and the dual . . . . . . . . . . . . . . . . . 452.4 Expressing an arbitrary element of V1I1 as a product of elements

of A = {ai : i I, , pi 6= 0}. . . . . . . . . . . . . . . . . . . 502.5 The connected graph (Q) . . . . . . . . . . . . . . . . . . . . . . 53

3.1 The graph (P ) and a neighbourhood of a collection of vertices. . 813.2 A square extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.3 A bipartite graph that has a symmetric distribution of edges with

respect to the perfect matching pi. . . . . . . . . . . . . . . . . . . 873.4 A symmetric distribution of idempotents in J3 . . . . . . . . . . . 893.5 Egg-box picture of the unique maximal J -class of the semigroup

I(2, 4, 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.1 Comparing the egg-box pictures of T3 and S2 o T3. . . . . . . . . . 1334.2 The structure of the unique maximal D-class of the semigroup

I(1, 2, 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.1 Comparing the egg-box pictures of the semigroup End(U) whereU = Z2 Z2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.2 The elements of the middle D-class of a monoid. . . . . . . . . . . 1525.3 The affine subspaces of V = Z3 Z3. . . . . . . . . . . . . . . . . 1565.4 Egg-box picture for End(Aff(V )). . . . . . . . . . . . . . . . . . . . 1605.5 Egg-box picture for End(Aff(V [+W ])). . . . . . . . . . . . . . . . . 164

6.1 The graphs associated with the maximal completely simple semi-groups in D2 T5. . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.2 The bipartite graphs corresponding to the maximal order com-pletely simple semigroups in D2. . . . . . . . . . . . . . . . . . . . 180

6.3 Two component bipartite graphs such that the largest componentsare maximal and the total number of edges is a power of two. . . 181

ix

7.1 Visualising the right boundary of the subsemigroup T of the semi-group S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.2 The right Cayley graph of the bicyclic monoid. . . . . . . . . . . . 1917.3 A diagram representing the ReidemeisterSchreier rewriting pro-

cess for semigroups. . . . . . . . . . . . . . . . . . . . . . . . . . . 1997.4 A patchwork quilt of semigroups. . . . . . . . . . . . . . . . . . . . 222

8.1 The right Cayley graph of S with respect to the generating sets Aand B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

x

Preface

This thesis is a collection of problems, and some solutions, from the theory ofsemigroups. It is divided into two parts. In the first part finite semigroups areconsidered and some combinatorial questions are investigated. In the second partinfinite semigroups are considered, in particular we look at a number of questionsin the theory of semigroup presentations. A summary of the topics covered inthis thesis is given below.

In Chapter 1 the necessary semigroup theory and graph theory preliminariesare given. In Chapter 2 minimal generating sets for finite completely 0-simplesemigroups are investigated. In particular, we give a formula for the minimumcardinality of a generating set for such a semigroup. Several applications to var-ious finite semigroups of transformations are also given. In Chapter 3 semibands(idempotent generated regular semigroups) are considered. Connections betweenminimal generating sets of idempotents of semibands, and matchings in bipartitegraphs are explored. Necessary and sufficient conditions for a completely 0-simplesemigroup to have an extremal idempotent generating set are given. These resultsare applied to ideals of the full transformation and the general linear semigroups.A consequence of this is a result giving necessary and sufficient conditions fora subset of a two-sided ideal of the general linear semigroup to be a minimalgenerating set.

Independence algebras and their endomorphism monoids are the subject ofChapters 4 and 5. The main results of these chapters generalise combinatorialresults of Howie and McFadden for the full transformation semigroup, and re-sults of Dawlings for the general linear semigroup, to the more general contextof endomorphism monoids of independence algebras. The necessary and suffi-cient conditions for completely 0-simple semigroups to have extremal idempotentgenerating sets, established in Chapter 3, provide the basis on which the mainresults of Chapters 4 and 5 are constructed. We finish our consideration of finitesemigroups in Chapter 6 where an extremal problem for subsemigroups of thefull transformation semigroup is discussed. The largest order completely simplesubsemigroups of the full transformation semigroup are described using an argu-ment that involves counting the number of distinct r-colourings of members of acertain family of r-partite graphs.

Chapter 7 is the place that we begin our study of infinite semigroups andin particular the theory of semigroup presentations. Given a finitely generatedsemigroup S and subsemigroup T of S we define the notion of the boundary of

xi

T in S which, intuitively, describes the position of T inside the left and rightCayley graphs of S. We prove that if S is finitely generated and T has a finiteboundary in S then T is finitely generated. We also prove that if S is finitelypresented and T has a finite boundary in S then T is finitely presented. Severalcorollaries and examples are given. In Chapter 8 we continue with the subject-matter introduced in Chapter 7. In particular, the boundaries of left and rightunitary subsemigroups are analysed which leads to results for presentations ofunitary subsemigroups. Several applications are given.

The main results of the thesis have been written up as a series of five researcharticles; see [42], [43], [44], [45], [46]. For all of the main results of this thesis,approaching the problem from the point of view of graphs proved to be the rightway of thinking about the problem. I hope the ideas that appear here will be ofinterest to those who read them and especially to others who, like me, think inpictures.

xii

Chapter 1

Preliminaries

1

2 Chapter 1, Preliminaries

1.1 Semigroup theory preliminaries

In this section all of the basic semigroup theory needed to understand the resultsof the thesis will be presented. All of the definitions and results are standard andcan be found in any introductory text on the subject (see for example [57], [64],[49], [52] or [21]).

Subsemigroups and generating sets

A semigroup is a pair (S, ) where S is a non-empty set and is a binary operationdefined on S that satisfies the associative law

(x y) z = x (y z)

for all x, y, z S. The product of two elements x and y is usually written just asxy rather than x y. If a semigroup contains an element 1 with the property thatx1 = 1x = x for all x S then we call 1 the identity element of the S and wecall S a monoid. If a semigroup contains an element 0 that satisfies x0 = 0x = 0for all x S then 0 is called a zero element of the semigroup. A semigroup canhave at most one identity element and at most one zero element.

We use S1 and S0 to denote the semigroup S with an identity or a zeroadjoined, respectively. That is,

S1 =

S if S has an identity elementS {1} otherwiseand

S0 =

S if S has a zero elementS {0} otherwise.If A and B are subsets of a semigroup we define the product of the two sets asAB = {ab : a A, b B}. In the special case of singleton subsets A = {a} wewrite aB rather than {a}B. So for example, S1a = Sa {a}.

A non-empty subset T of S is called a subsemigroup if it is closed undermultiplication. Let S be a semigroup and let {Ti : i I} be an indexed set ofsubsemigroups of S. Then if the set

iI Ti is non-empty, it is a subsemigroup

of S. In particular, for any non-empty subset A of S the intersection of all thesubsemigroups of S that contain A is non-empty and is a subsemigroup of S. Weuse A to denote this subsemigroup and call it the subsemigroup of S generated

Section 1.1 3

by the set A. The subsemigroup A is the set of all elements in S that can bewritten as a finite product of elements of A.

An element e S is called an idempotent if it satisfies e2 = e. We use E(S)and F (S) to denote the idempotents of S and the subsemigroup generated by theset of idempotents, respectively. A band is a semigroup such that every elementis an idempotent.

Homomorphisms and congruences

A right congruence on a semigroup S is an equivalence relation that is stableunder multiplication on the right. In other words, for all a, s, t S

(s, t) (sa, ta) .

An equivalence relation which is stable under left multiplication is called a leftcongruence and a relation that is both a left and right congruence is called a(two-sided) congruence on S. If is a congruence on S then we can define abinary operation on the quotient set S/ by

(a)(b) = (ab).

A map : S T where S and T are semigroups is called a homomorphismif for all x, y S

(xy) = xy.

If S and T are monoids then, to be called a monoid homomorphism, must alsosatisfy 1S = 1T . A homomorphism that is injective will be called a monomor-phism and if it is surjective it will be called an epimorphism. Also, a homomor-phism is called an isomorphism if it is bijective. When there exists an epimor-phism from S onto T we say that T is a homomorphic image of S. If there is anisomorphism : S T we say that S and T are isomorphic and write S = T . Ahomomorphism from S to itself is called an endomorphism and an isomorphismfrom S to itself is called an automorphism. The set of all endomorphisms of S,under composition of maps, forms a monoid. We call this monoid the endomor-phism monoid of S and denote it by End(S). Similarly, the set of automorphismsforms a group that is denoted Aut(S) and is called the automorphism group ofS. Given a map : S T we define

ker = {(x, y) S S : x = y}


and call this the kernel of the map . The first isomorphism theorem for semi-groups tells us that with every epimorphism : S T the kernel ker is acongruence on S and S/ ker = T . Conversely, if is a congruence on S thenthe map : S S/ defined by x = x/ is an epimorphism from S onto thefactor semigroup S/.

Ideals and Rees quotients

A subsemigroup T of a semigroup S that satisfies TS T is called a right ideal.Dually, if ST T then T is called a left ideal and T is called a (two-sided) idealif it is both a left and a right ideal. An ideal I of S is called proper if I 6= S. IfI is a proper ideal of a semigroup S then

I = {(s, s) : s S} (I I)

is a congruence on S. It is useful to think of S/I as (S \ I) {0} where allproducts not falling in S \ I are equal to zero. We shall call a congruence of thistype a Rees congruence, and if a homomorphism : S T is such that ker is aRees congruence we shall say that is a Rees homomorphism. We shall normallywrite S/I rather than S/I and call this the Rees quotient of S with respect to I.

Regular semigroups, Greens relations and the structure of a D-class

Greens relation were first introduced in [48]. They describe the ideal structureof a semigroup. Since their introduction they have played a central role in thestructure theory of semigroups. We now define Greens relations R, L, J , D andH and give some of their basic properties.

Let S be a semigroup and let s S. The principal right, left and two-sided ideals generated by s are the sets sS1 = sS {s}, S1s = Ss {s} andS1sS1 = sS Ss SsS {s}, respectively. For s, t S we say that s and t areR-related, writing sRt, if s and t generate the same principal right ideal. We saythey are L-related if they generate the same principal left ideal, in which case wewrite sLt. Also, we say s and t are J -related, writing sJ t, if they generate thesame principal two-sided ideal. We define H = R L and D = R L = L R:the composition of the binary relations R and L. Each of these relations is anequivalence relation on the semigroup S and we call the corresponding equivalenceclasses the R-, L-, J -, D- and H-classes, respectively of S.

Given an element s S, we will use Rs, Ls, Js, Ds and Hs to denote the R-,

Section 1.1 5

L-, J -, D- and H-classes, respectively, of s in S. Since R, L and J are definedin terms of ideals, ordering the right, left and two-sided ideals of S by inclusioninduces a partial order on these equivalence classes given by

La Lb S1a S1bRa Rb aS1 bS1Ja Jb S1aS1 S1bS1.

In finite semigroups the relations D and J coincide and, when working with finitesemigroups, we write Ds Dt to mean Js Jt.

Each D-class of a semigroup S is a union of R-classes and also is a union ofL-classes. Moreover, aDb if and only if Ra Lb 6= which is true if and only ifRb La 6= . If D is a D-class of S and if a, b D are R-related in S, say withas = b and bs = a, then the right translation s : S S defined by xs = xsmaps La to Lb. The map s : S S maps Lb back to La and the composition ofthe maps ss : S S acts as the identity map on La. Moreover, the map s isR-class preserving in the sense that it maps each H-class on La in a 1-1 manneronto the corresponding (R-equivalent) H-class of Lb. There is a dual result forL-classes. These results, collectively, are known as Greens lemma.

It is often useful to visualise a D-class of a semigroup using a so called egg-boxdiagram. An egg-box diagram of a D-class D is a grid whose rows represent theR-classes of D, its columns represent the L-classes of D, and the intersectionsof the rows and columns, that is, the cells of the grid, represent the H-classes ofthe semigroup. Egg-box diagrams will be found scattered amongst the chaptersof this thesis. They provide a useful tool for visualising semigroups.

The H-classes of a given D-class all have the same size. Each H-class H of Sis either a subgroup of S or satisfies H2 H = . An H-class is a subgroup ofS if and only if it contains an idempotent (which will act as the identity of thatsubgroup). We call the H-classes that contain idempotents the group H-classesof S. Any two group H-classes in a given D-class are isomorphic.

An element a S is called regular if there exists x S such that axa = a.The semigroup S is said to be regular if all of its elements are regular. If Dis a D-class then either every element in D is regular or none of them are. TheD-classes that have regular elements are called the regular D-classes. In a regularD-class each R-class and each L-class contains an idempotent. If a S then wesay that a is an inverse of a if

aaa = a, aaa = a.


An element has an inverse if and only if that element is regular. The followinglemma is used extensively in the thesis.

Lemma 1.1. Let a, b be elements in a D-class D. Then ab Ra Lb if andonly if La Rb contains an idempotent.

0-simple semigroups, the Rees theorem and principal factors

A semigroup is called simple if it has no proper ideals. This is equivalent tosaying that the semigroup has a single J -class. A completely simple semigroupis a simple semigroup that has minimal left and right ideals. Every finite simplesemigroup is completely simple.

A left zero semigroup is a semigroup in which every element acts as a left zero(i.e. in which xy = x for all x, y S). A right zero semigroup is one in whichevery element acts as a right zero. Note that left and right zero semigroups arejust special kinds of completely simple semigroup.

A semigroup is called 0-simple if {0} and S are its only ideals (and S2 6= {0}).This is equivalent to saying that {0} and S \ {0} are its only J -classes (andS2 6= {0}). A semigroup is 0-simple if and only if SaS = S for every a 6= 0 in S.A semigroup S is said to be completely 0-simple if it is 0-simple and has 0-minimalleft and right ideals. By a 0-minimal left (respectively right) ideal we mean aleft (respectively right) ideal that is minimal within the set of all non-zero left(respectively right) ideals ordered by inclusion. Every finite 0-simple semigroupis completely 0-simple.

0-simple semigroups occur naturally inside arbitrary semigroups appearingas principal factors of J -classes. Let J be some J -class of a semigroup S. Thenthe principal factor of S corresponding to J is the set J = J {0} with multi-plication

s t ={

st if s, t, st J0 otherwise.

The semigroup J is either a semigroup with zero multiplication or is a 0-simplesemigroup. The following construction, due to Rees, gives a method for buildingcompletely 0-simple semigroups

Let G be a group, let I, be non-empty index sets and P = (pi) a regularI matrix over G{0} (where regular means that every row and every columnof P has at least one non-zero entry). Then S = M0[G; I,;P ], the I Rees matrix semigroup over the 0-group G {0} with sandwich matrix P , is the

Section 1.1 7

semigroup (I G ) {0} with multiplication defined by

(i, g, )(j, h, ) =

{(i, gpjh, ) if pj 6= 00 otherwise

(i, g, )0 = 0(i, g, ) = 00 = 0.

The semigroup S = M0[G; I,;P ] is completely 0-simple. Moreover, by [57,Theorem 3.2.3] every completely 0-simple semigroup is isomorphic to someM0[G; I,;P ].

Semigroups of transformations

LetX be a non-empty set. The symmetric group SX consists of all bijections fromX to itself under composition of maps. The full transformation semigroup TXconsists of all maps from X into X under the operation of composition of maps.The partial transformation semigroup PX consists of all partial maps of X, whilethe symmetric inverse semigroup IX consists of all partial one-one maps of X.When |X| = n we often write Sn, Tn, Pn and In in place of SX , TX , PX and IXrespectively. When |X| = n we often identify X with the set Xn = {1, 2, . . . , n}.For every TX we define

im = {x : a X}

and call this the image of the map . Also we define

ker = {(x, y) X X : x = y}

and call this the kernel of the map . This is clearly an equivalence relation onthe set X. We call the equivalence classes of ker the kernel classes of the map. A partition of the set X is a family of pairwise disjoint, non-empty subsetsof X whose union is X. Thus the kernel classes of a transformation in TX area partition of the set X. Often it will be convenient to write ker in terms ofthis partition rather than as a subset of X X. For example, the kernel of theelement

=

(1 2 3 41 2 2 3

) T4

would be written as ker = {{1}, {2, 3}, {4}}.Given a partition iIXi of X we define the weight of the partition to be

|I|. The Stirling number of the second kind, S(n, r), is the number of partitions


of {1, . . . , n} with r (non-empty) parts. This number is given by the followingrecursion formula:

S(n+ 1, k) = kS(n, k) + S(n, k 1)

where S(1, 0) = 0 and S(1, 1) = 1. Given a partition of the set X we call a systemof distinct representatives of this family of sets a transversal of the family.

Direct and semidirect products

If S and T are semigroups then the set S T with multiplication

(s1, t2)(s2, t2) = (s1s2, t1t2)

forms a semigroup that we call the direct product of S and T . More generally,let T and S be semigroups and be a homomorphism of S into End(T ). Denoteby st the value of (s) End(T ) at t T . We view as a left action of S onT , s1(s2t) = s1s2t, by endomorphisms s(t1t2) = st1st2. The semidirect product ofT and S over is the semigroup T S on the Cartesian product T S withmultiplication

(t1, s1)(t2, s2) = (t1s1t2, s1s2).

In the special case where (S) = {1End(T )} End(T ) the semidirect product ofT and S over is equal to the direct product of S and T .

Free semigroups, monoids and presentations

Let A be a non-empty set. Let A+ be the set of all finite, non-empty words inthe alphabet A. With respect to the binary operation of juxtaposition of wordsthe set A+ forms a semigroup that we call the free semigroup on A. The set Ais a generating set for A+ and it is the unique minimal generating set of A+.Adjoining an identity 1 to the free semigroup A+ gives the free monoid which wedenote A. We think of the identity of the free semigroup A as the empty wordand sometimes denote it by . Every semigroup can be expressed as a quotientof a free semigroup by a congruence. If A is a finite alphabet and if we can find afinite set R A+ A+ such that S = A+/, where is the smallest congruenceof A+ containing R, then we say that S is finitely presented.

Section 1.2 9

1.2 Graph theory preliminaries

We need some basic concepts from graph theory. The ideas presented here arestandard and may be found in any introductory text on graph theory. See forexample [51], [10], [8] and [15].

Subgraphs and isomorphisms

A graph is a pair (V, E) where V is a set and E is a set of 2-subsets of V. Theset V is the set of vertices and the set E is the set of edges. Given a graph weuse V() and E() to denote the set of vertices and the set of edges, respectively,of the graph . An edge {i, j} is said to join the vertices i and j and this edgeis denoted ij. The vertices i and j are called the endvertices of the edge ij. Ifij E() we say that the vertices i and j are adjacent in the graph . We saythat i and j are incident to the edge ij. We say that two edges are adjacent ifthey have a common incident vertex.

We think of a graph as a collection of vertices, some of which are joined byedges, and as a result graphs are often represented as pictures. For example, thegraph = (V, E) = ({1, 2, 3, 4}, {{1, 2}, {2, 3}, {3, 4}}) is given in Figure 1.1.

The graph = (V , E ) is a subgraph of if V V and E E . If containsall the edges of that join vertices of V then we call the subgraph of inducedby V .

Given a subsetW of V() we use W to denote the subgraph of obtainedby deleting the vertices W and all of the edges adjacent with them. Similarly,given a subset F of the edge set E() we use F to denote the subgraphobtained by deleting the edges F . An elementary contraction of a graph isobtained by identifying two adjacent vertices u and v, that is, by deleting u andv and replacing them by a single vertex w adjacent all to the vertices to which u orv were adjacent. A graph is contractible to a graph if can be obtained from by a finite sequence of elementary contractions. If the graph is contractableto the graph then we call a contraction of .

The graphs = (V, E) and = (V , E ) are said to be isomorphic if there

2 3

14

Figure 1.1: A graph.


is a bijection : V V that preserves adjacency. That is, for all x, y V,xy E if and only if (x)(y) E . We write = to mean that and areisomorphic.

The complete graph of order n is defined to be the unique graph with n verticesand

(n2

)edges, so that there is an edge connecting every pair of vertices, and is

denoted Kn.The degree of a vertex v is the number of edges adjacent to it and will be

denoted by d(v). The set of vertices adjacent to a vertex v is called the neigh-bourhood of v and is denoted N(v). More generally, if W is a subset of V thenthe neighbourhood of W, denoted N(W), is defined to be the set of all verticesof V \ W which are neighbours of at least one vertex from W. We call a graphk-regular if every vertex has degree k for some number k. A graph is regular ifit is k-regular for some k.

Paths and connectedness

A path in a graph is a set of vertices pi = {v0, v1, . . . , vn} such that vi1vi belongto E() for all 1 i n. We call v0 and vk the initial and terminal vertices,respectively, of the path pi. A walk is a sequence (v0, e1, v1, e2, v2, . . . , en, vn)where ei is the edge vi1vi for i = 1, . . . , n. A trail is a walk where all the edgesare distinct. A trail whose endvertices coincide is called a circuit. A walk with atleast three vertices, where all the vertices are distinct, and where the endverticescoincide, is called a cycle.

A graph is called connected if for every pair {x, y} of distinct vertices thereis a path from x to y. The maximum connected subgraphs of a graph are calledthe connected components of . A forest is a graph with no cycles and a tree is aconnected graph with no cycles. Therefore, the connected components of a forestare all trees. A spanning tree of a graph is a subgraph T that is a tree and thatcontains every vertex of the graph . Frequent use will be made of the followingeasy result.

Lemma 1.2. Every connected graph contains a spanning tree.

Of course, if has n vertices and T is a spanning tree of then T has nvertices and n 1 edges.

A graph is bipartite if V can be written as the disjoint union of two sets V1and V2 in such a way that every edge in E has one vertex in V1 and the otherin V2. We say that the bipartite graph = V1 V2 is balanced if |V1| = |V2|.Similarly, we say that the graph is r-partite with vertex classes V1, . . . ,Vr if

Section 1.2 11

V () = V1 . . . Vr, and Vi Vj = whenever i 6= j and no edge joins twovertices of the same class. A complete r-partite graph is denoted Kn1,...,nr , ithas ni vertices in the ith class and contains all edges joining vertices in distinctclasses.

In the above definition of graph we do not allow multiple edges or loops (anedge joining a vertex to itself). We call a graph that is allowed multiple edgesand loops a multigraph. If the edges are ordered pairs (rather than two-sets) thenwe get the notion of a digraph (directed graph) and directed multigraph. Thenotions above for graphs, such as paths and walks, carry over to the context ofmultigraphs and digraphs in a natural way.

Hamiltonian graphs

A cycle containing all the vertices of a graph is said to be a Hamiltonian cycle.A Hamiltonian path is a path containing all the vertices of a graph. A graphcontaining a Hamiltonian cycle is said to be Hamiltonian. No efficient algorithmis known for constructing a Hamiltonian cycle, though neither is it known that nosuch algorithm exists. On the other hand, some sufficient conditions for a graphto be Hamiltonian are known. The following result gives a sufficient conditionfor a bipartite graph to be Hamiltonian.

Theorem 1.3 (Moon and Moser, [75]). If = X Y is a bipartite graph with|X| = |Y | = n such that any non-adjacent pair of vertices (x, y) XY satisfiesd(x) + d(y) n+ 1, then is Hamiltonian.

Matchings and Halls theorem

A subset F of E() is called independent if no two edges have a vertex in common.Similarly, a subset V of V() is called independent of no two vertices in V areadjacent. Amatching in a graph is a set of independent edges. A perfect matchingis a matching on |V|/2 edges. In particular, in a bipartite graph = A Bassociated with any perfect matching is a bijection pi : A B that satisfies{x, xpi} E() for all x A. Note that if = AB has a perfect matching then|A| = |B|.

Let A1, . . . , An be sets. A system of distinct representatives (SDR) for thesesets is an n-tuple (x1, . . . , xn) of element with the properties:

(i) xi Ai for i = 1, . . . , n;

(ii) xi 6= xj for i 6= j.


Halls marriage theorem gives necessary and sufficient conditions for a family(A1, . . . , An) of finite sets to have a SDR.

Theorem 1.4 (Halls theorem, [50]). The family (A1, . . . , An) of finite sets tohave a SDR if and only if

jJ

Aj |J | for every J {1, . . . , n}. (1.1)

When a family of sets (A1, . . . , An) satisfies condition (1.1) we say that itsatisfies Halls condition. A family A = (A1, . . . , An), where Ai X for all i,is naturally identifiable with a bipartite graph with vertex classes V1 = A andV2 = X where Ai A is joined to x X if and only if x Ai. A system ofdistinct representatives is then just a perfect matching in this bipartite graph. Inthis context Halls marriage theorem becomes.

Theorem 1.5. The bipartite graph G = X Y has a perfect matching if andonly if |N(A)| |A| for all subsets A of X.

A colouring of a graph is an assignment of colours to the vertices such thatadjacent vertices have distinct colours. A k-colouring of is a function c : V(){1, 2, . . . , k} such that for each j the set c1(j) is independent. The chromaticnumber () of the graph is the minimal number of colours in a vertex colouringof the graph .

The dual of the graph is the graph D() with vertex set V() and ij E(D()) if and only if ij 6 E().

Part I

Finite semigroup theory

13

Chapter 2

Generating sets for completely

0-simple semigroups using

bipartite graphs

15

16 Chapter 2, Minimal generating sets using bipartite graphs

2.1 Finite semigroups and their generating sets

It is often convenient to give a finite semigroup S in terms of a set of generators A.In many cases this set may be chosen to have considerably fewer elements than Sitself. For example, the full transformation semigroup Tn has nn elements while itmay be generated by just three transformations. In particular, the transposition(1 2), the n-cycle (1 2 . . . n), and any transformation satisfying | im| = n1,together will generate Tn (see [57, Exercise 1.7]).

In this chapter we will be concerned with the problem of finding smallgenerating sets for finite semigroups. Given a semigroup S we will use rank(S)to denote the minimum cardinality of a generating set for S. In other words:

rank(S) = min{|A| : A = S}.

Our interest is in determining rank(S) and, whenever possible, in describing allgenerating sets with this size. We will call any generating set of S with sizerank(S) a basis of the semigroup S.

The ranks of a wide number of finite groups are well known. In a finite groupG every subsemigroup is a subgroup (since for any g G we have gm = g1for some positive number m) and thus, given a subset A of G, the subsemigroupof G generated by A is equal to the subgroup of G generated by A. There is,therefore, no distinction between the group rank and the semigroup rank of G.For infinite groups this is not necessarily the case. For example, the infinite cyclicgroup Z has rank 2 as a semigroup but rank 1 as a group. It is well known thatthe symmetric group satisfies rank(Sn) = 2 for n 3, as does the alternatinggroup An for n 4. In fact, any finite non-abelian simple group G has rank2. This result is a consequence of the classification of finite simple groups. Therank of any non-trivial finite general linear group is also known to equal 2 (seefor example [94]).

The function rank : S N, from the class (pseudo-variety) of all finitesemigroups to the natural numbers, does not behave well with respect to takingsubsemigroups. For example, by Cayleys theorem, every finite semigroup isembeddable in some finite full transformation semigroup Tn while, as alreadymentioned, rank(Tn) = 3 for all n 3. On the other hand, if T is the image of Sunder a homomorphism then it is clear that rank(S) rank(T ). This is becauseof the following simple observation.

Lemma 2.1. Let S and T be semigroups, let A be a subset of S and let :S T be an epimorphism. If A generates S then A generates T . In particular

Section 2.1 17

|A| |A| and rank(S) rank(T ).

In [23] a group G is defined to be generator critical if all of its proper homo-morphic images H satisfy rank(G) > rank(H). The idea being that if a groupis not generator critical then we may factor down to an easier group with thesame rank. Exactly the same idea carries over to semigroups and the general ideaof studying large homomorphic images of S in order to determine its rank willbe a re-occurring theme throughout this chapter.

In terms of semigroup theory, the question of rank has been considered mainlyfor various semigroups of transformations. The theory of transformation semi-groups is one of the oldest and most developed within semigroup theory. In [58]Howie argues that

It is this connection with maps (arising from the associative axiom)that is the strongest reason why semigroups are more important boththeoretically and in applications that the various non-associative gen-eralizations of groups.

Early work on generators and relations in transformation semigroups wascarried out by Azenstat in [2] and [3]. In [37] Gomes and Howie prove thatthe semigroup Singn of of all singular self-maps of Xn satisfies rank(Singn) =n(n 1)/2. In the same paper they also consider the semigroup SPn In,of all proper subpermutations of Xn, proving that rank(SPn) = n + 1. In [59]Howie and McFadden generalized the above result for the semigroup of singularmappings by considering a general two-sided ideal of Tn. These ideals have theform:

K(n, r) = { Tn : | im| r}

where 1 r n (we will see the reason for this in Proposition 2.14). In particular,in [59] it is shown that rank(K(n, r)) = S(n, r): the Stirling number of the secondkind. Garba, in [34], considered the semigroup of all partial transformations Pnon the set Xn and showed rank(KP (n, r)) = S(n+ 1, r + 1) where

KP (n, r) = { Pn : | im| r}.

In [36] he also generalised Gomes and Howies result for SPn by showingrank(L(n, r)) =

(nr

)+ 1, where

L(n, r) = { In : | im| r}.


Various order preserving versions of the examples above have also been consid-ered. Originally in [4] Azenstat considered the semigroup of order preservingtransformations:

On = { Singn : (x, y Xn) x y x y}.

She showed that it is idempotent generated and that it has a uniquely determinedirreducible set of idempotent generators (namely the identity along with all theidempotents e that satisfy | im e| = n1). This result was later reproven by Howiein [38]. We will see more on idempotent generating sets in subsequent chapters.Also, in [38] it was shown that the semigroup On has rank n. In the same paperthe semigroup of partial order preserving transformations of Xn (excluding theidentity map):

POn = On { : dom() ( Xn, (x, y dom()) x y x y}

was shown to have rank 2n 1 and the strictly partial order preserving transfor-mations:

SPOn = POn \Onwere shown to have rank 2n2. Also, in a series of papers [68], [67] and [66] Leviand Seif have considered semigroups generated by transformations of prescribedpartition type. These semigroups are closely related to the Sn-normal semigroupsintroduced in [65].

Given an arbitrary finite semigroup S, if A generates S and JM is somemaximal J -class of S then A JM must generate the principal factor JM . Asa consequence, the rank of S is equal to at least the sum of the ranks of theprincipal factors that correspond to the maximal J -classes of S. If S happens tobe generated by the elements of its maximal J -classes then rank(S) is preciselyequal to this sum. In fact, this is a property that is shared by the majorityof the semigroups described above. As a consequence, in each case the rank ofthe semigroup in question is equal to the rank of a corresponding completely 0-simple semigroup. In this way, these results act as motivation for finding a generalformula for the rank of an arbitrary finite completely 0-simple semigroup.

The first occurrence of a formula for the rank of a completely 0-simple semi-group can be found in [37] where, in order to find the rank of the semigroup SPn,the authors give an expression for the rank of an arbitrary Brandt semigroupB(G, {1, . . . , n}) in terms of its dimension n, and of the rank of the underlyinggroup G. In another paper [81] the author considers a class of completely 0-simple

Section 2.2 19

semigroups he calls connected, a restriction on the form of the matrix P whichin particular is satisfied by all completely simple semigroups, and gives a formulafor the rank of an arbitrary connected completely 0-simple semigroup.

In this chapter we will build on the ideas of [81] giving a general formula forthe rank of an arbitrary completely 0-simple semigroup in terms of the groupG, the size of the index sets I and , the number of components in the ma-trix P , and a special term rmin that will be defined. In 2.2 some preliminaryresults are introduced then in 2.3 the special case of combinatorial completely0-simple semigroups (those whose maximal subgroups are trivial) is considered.Graph theoretic methods for working with completely 0-simple semigroups areintroduced in 2.4. Results for connected completely 0-simple semigroups aregiven in 2.5 and in 2.62.9 the general case is considered and the main resultsof the chapter are presented. A normalization theorem is the subject of 2.10and finally, in 2.112.13, several applications of the main results are discussed.

2.2 Preliminaries

Let S = M0[G; I,;P ] be a finite completely 0-simple semigroup. We will useRi, L and Hi to denote the R, L and H-classes indexed by i I, and(i, ) I respectively. We can only hope to answer our question modulogroups and the concept of relative rank gives us a way of accomplishing this.Given a subset A of a semigroup S, we define the relative rank of S modulo Aas the minimum number of elements of S that need to be added to A in order togenerate the whole of S:

rank(S : A) = min{|X| : A X = S}.

Example 2.2. From the discussion at the beginning of Section 2.1 we concludethat rank(Tn : Sn) = 1.

Since all the semigroups we consider here have a zero we will always includethe zero in any given subsemigroup. As a consequence of this by X we willmean all the elements that can be written as products of elements of X, pluszero if necessary. This is really just a matter of convenience and we do not loseanything by doing it. Without this convention we would have to deal with thecases 0 (S \ {0})2 and 0 6 (S \ {0})2 separately with the rank differing by 1each time.

The following lemma states the obvious fact that a generating set for S =M0[G; I,;P ] must intersect every R- and every L-class of S.


Lemma 2.3. Let S =M0[G; I,;P ] be a completely 0-simple semigroup. Thenrank(S) max(|I|, ||).

As a direct consequence of Greens Lemmas (see Chapter 1) it follows that ifA generates every element of a single group H-class and at least one element inevery other H-class then this is enough to say that A generates the whole of S.More precisely:

Lemma 2.4. [81, Lemma 3.7] Let S =M0[G; I,;P ] be a completely 0-simplesemigroup, let Hi be a group and let A S. If Hi A and A Hj 6= for all j I, then S = A.

The result above may be thought of as analogous to the following situationin group theory. Let G be a group and let N be a normal subgroup of G. LetA be a subset of G. If A N and A/N = G/N then A = G. Here N isplaying the role of Hi and the cosets of N in G play the role of the non-trivialH-classes of S.

Roughly speaking, if S is a completely 0-simple semigroup and A is a gener-ating set for S then every generator a A makes a two-fold contribution. Firstly,the generator contributes to generating at least one element in every H-class ofS. Secondly, each generator contributes to generating the underlying group.

2.3 Rectangular 0-bands

A rectangular 0-band, denoted by S =M0[{1}; I,;P ], is a 0-Rees matrix semi-group over the trivial group. Understanding the generating sets of rectangular0-bands will give us a useful first step towards understanding generating sets ofRees matrix semigroups over non trivial groups. Since the middle component ofevery triple equals 1 we can effectively ignore it and consider the semigroup ofpairs S = (I ){0} with I = {1, 2, . . . ,m} and = {1, 2, . . . , n}, P a regularnm matrix over {0, 1}, whose multiplication is given by

(i, )(j, ) =

{(i, ) if pj = 10 if pj = 0

(i, )0 = 0(i, ) = 00 = 0.

Associated with any completely 0-simple semigroup S =M0[G; I,;P ] is therectangular 0-band given by replacing all the non-zero entries in the matrix P bythe symbol 1 and by replacing G by the trivial group.

Section 2.3 21

Definition 2.5. Given S = M0[G; I,;P ] by the natural rectangular 0-bandhomomorphic image of S we mean S/H. This semigroup can be concretely rep-resented as T =M0[{1}; I,;Q] where qi = 1 if pi 6= 0, and qi = 0 otherwise.We will use \ to denote the corresponding epimorphism from S to T with 0\ = 0and (i, g, )\ = (i, ).

We call two nm matrices A and B over {0, 1} equivalent, writing A B, ifB can be obtained from A by permuting its rows and columns. Clearly is anequivalence relation on the set of nm matrices over {0, 1} and, as a special caseof Theorem 2.54 below, two matrices are equivalent if and only if the rectangular0-bands that they correspond to are isomorphic.

Definition 2.6. Let P be an n m matrix over {0, 1}. Then we useP [i1, . . . , il][j1, . . . , jk] to denote the submatrix of P obtained by deleting all ele-ments with first coordinate in the set {i1, . . . , il} or second coordinate in the set{j1, . . . , jk}. We use 0 when no rows or columns are to be deleted. For exampleP [0][1] means leave the rows alone but delete column 1.

First we will show that given a non-square matrix we can always delete a rowor a column while maintaining regularity.

Lemma 2.7. Let P be a regular nm matrix over {0, 1}.

(i) If m > n then there exists j {1, . . . ,m} such that Q = P [0][j] is regular.

(ii) If m < n then there exists i {1, . . . , n} such that Q = P [i][0] is regular.

(iii) If n = m then there exists j {1, . . . , n} such that P [0][j] is regular if andonly if there exists i {1, . . . , n} such that P [i][0] is regular which is thecase if and only if P 6 In (the n n identity matrix).

Proof. (i) Suppose otherwise, so that for all j {1, . . . ,m} the matrix P [0][j] isnot regular. Then for each j {1, . . . ,m} there is a row that has 1 in the jthposition and zeros everywhere else. All of these rows are distinct and there are mof them which contradicts the fact that m > n. (ii) Use a symmetric argumentto that of part (i). (iii) By the same argument as in part (i) it follows that ifP [0][i] is not regular for all i {1, . . . , n} then P In. Conversely, if P In itis clear that for any i {1, . . . , n}, P [0][i] is not regular.

Corollary 2.8. Let P be a regular n n matrix over {0, 1} such that P 6 In.Then there exist i, j {1, . . . , n} such that P [0][j], P [i][0] and P [i][j] are allregular.


Proof. By Lemma 2.7 we can find i, j {1, . . . , n} so that P [0][j] and P [i][0] areregular. Hence P [i][j], which is their intersection, is clearly regular.

The Brandt semigroup B = B(G, {1, . . . , n}) is the Rees matrix semigroupM0[G; I, I;P ] where P In, the n n identity matrix, and I = {1, . . . , n}.When G is the trivial group B(G, I) is a rectangular 0-band that we call theaperiodic Brandt semigroup and denote by Bn.

Lemma 2.9. The aperiodic Brandt semigroup Bn has rank n.

Proof. () By Lemma 2.3 rank(Bn) n. () The set {(1, 2), (2, 3), . . . , (n 1, n), (n, 1)} generates Bn. Indeed, if (x, y) Bn then:

(x, y) =

(x, x+ 1)(x+ 1, x+ 2) . . . (y 1, y) if y > x(x, x+ 1)(x+ 1, x+ 2) . . . (n 1, n)(n, 1)(1, 2) . . . (y 1, y) if y x.

In fact, it is fairly easy to describe all the bases of the aperiodic Brandtsemigroup Bn and we will do so at the end of 2.4. Corollary 2.8 forms the basisof the inductive step that proves the following result.

Theorem 2.10. Let S =M0[{1}; I,;P ] be an mn rectangular 0-band. Then

rank(S) = max(m,n).

Proof. It follows from Lemma 2.3 that rank(S) max(m,n). Now we have toshow that we can always find a generating set of this size. We must prove:

U(m,n) : If S is an m n rectangular 0-band then S has a generatingset with cardinality max(m,n).

First we consider the case where m = n and use induction on n. U(1, 1) holdstrivially. Suppose U(k, k) holds and let T be a (k + 1) (k + 1) rectangular0-band with I = = {1, 2, . . . , k + 1}, and underlying matrix P . If P Ik+1then U(k + 1, k + 1) holds by Lemma 2.9. If P 6 Ik+1 then by Corollary 2.8we can suppose without loss of generality that the submatrices M = P [k + 1][0],N = P [0][k + 1] and O = P [k + 1][k + 1] are all regular. Let TM , TN and TObe the sub-rectangular 0-bands corresponding to these regular matrices. By theinductive hypothesis we can find A TO such that |A| = k and A = TO. Now

Section 2.3 23

let B = A {(k + 1, k + 1)} T . Clearly |B| = k + 1 and we also claim thatB = T . Indeed, we have

B = A {(k + 1, k + 1)} = A {(k + 1, k + 1)} = T0 {(k + 1, k + 1)}.

We are left to show {(k+1, i), (i, k+1) : i {1, . . . , k}} B. Let j {1, . . . , k}.By the regularity of M and N we can find v, l {1, . . . , k} such that p(k+1),l =pv,(k+1) = 1. Then we have

(k + 1, j) = (k + 1, k + 1)(l, j), (j, k + 1) = (j, v)(k + 1, k + 1)

with (l, j), (j, v) T0 which completes the inductive step.Now we consider the case where T is an m n rectangular 0-band with,

say, n > m. By repeated application of Lemma 2.7 without loss of generalitywe can suppose that Q = P [m + 1,m + 2, . . . , n][0] is regular. Let TQ be thesub-rectangular 0-band corresponding to Q. Since Q is an m m matrix bythe previous case we can find A TQ with |A| = m and A = TQ. Now letR = {(1, ) : = m + 1, . . . , n} and B = A R T . Clearly |B| = n and wealso claim that B = T . Indeed, we have

B = A R = A R = TQ R.

We are left to show {(i, j) : i {1, . . . ,m}, j {m + 1, . . . , n}} B. By theregularity of Q we can find x {1, . . . ,m} such that px1 = 1 and we concludethat (i, j) = (i, x)(1, j) where (i, x) TQ and (1, j) R.

Using just this simple result we may now determine the rank of a special classof completely 0-simple semigroup.

Definition 2.11. A semigroup S is called idempotent generated if E(S) = S.

Lemma 2.12. Let S be a finite idempotent generated semigroup and let A bea subset of S. If A has non-trivial intersection with every H-class of S then Agenerates S.

Proof. Since S = E(S) it is sufficient to prove E(S) A. Let e E(S) andlet b A He. Since He is a finite group with identity e it follows that bi = efor some i N, the smallest such i just being the order of b in the group He. Itfollows that e = bi A and since e was arbitrary that E(S) A.

Combining this with Theorem 2.10 gives:


Theorem 2.13. Let S = M0[G; I,;P ] be a finite completely 0-simple semi-group. If S is idempotent generated then rank(S) = max(|I|, ||).

Proof. The fact that rank(S) max(|I|, ||) follows from Lemma 2.3. For theconverse let T = S\, the natural rectangular 0-band homomorphic image of S.By Theorem 2.10 we can find a generating set A for T with size max(|I|, ||).The pre-image of A under the map \ is a union of H-classes of S. Let B be atransversal of this set of H-classes. Since A = T it follows that B has non-trivial intersection with every (non-zero) H-class of S and so B generates S, byLemma 2.12, and |B| = |A| = max(|I|, ||).

This result is not quite as obscure as it might at first seem. Many naturallyoccurring semigroups are idempotent generated and when, in addition to this,they are generated by the elements in their maximal J -classes, determining theirrank just reduces to the problem of counting the number of R- and L-classes.The full transformation semigroup Tn provides us with a family of examples ofthis kind.

Proposition 2.14. Let n N and let 1 r < n. Let , Tn.

(i) Greens relations are given by:

(a) R im = im;(b) L ker = ker;(c) D | im| = | im|.

(ii) K(n, r) = E(Dr).

(iii) The number of R-classes in Dr = { Tn : | im| = r} is the Stirlingnumber of the second kind S(n, r).

(iv) The number of L-classes in Dr is(nr

).

(v) The H-class indexed by the image I and the kernel K is a group if and onlyif I is a transversal of K.

Proof. (i) See [57, Exercise 1.16]. (ii) By [57, Theorem 6.3.1] the semigroup Singnis regular and idempotent generated. By [57, Exercise 6.12] if S is a semibandthen every element a S is expressible as a product of idempotents in Ja. Also,from [57, Lemma 6.3.2] it follows that Dr = K(n, r). The result is an immediateconsequence of these three facts. (iii) and (iv) are immediate consequences of (i).(v) See [57, Exercise 1.18].

Section 2.4 25

These facts, along with Theorem 2.13 allow us to determine the rank of thesemigroup K(n, r). This result was originally proven in [59, Theorem 5], wherethey also determined the so called idempotent rank of the semigroup. We willsee more about this in the following chapter.

Theorem 2.15. Let n N and let 1 < r < n. Then:

rank(K(n, r)) = S(n, r).

Proof. It follows from Proposition 2.14 that rank(S) = rank(K(n, r)/K(n, r1))and that K(n, r)/K(n, r 1) is idempotent generated. Applying Theorem 2.13to the idempotent generated completely 0-simple semigroup K(n, r)/K(n, r 1)gives:

rank(K(n, r)) = rank(K(n, r)/K(n, r 1)) = max(S(n, r),(n

r

)) = S(n, r).

Theorem 2.13 may be applied to a number of other examples. In particularthe exact analogue of the above result may be proven for the ideals of End(V )where V is a finite vector space. See Section 3.5.2 for more details on this.

Another consequence of Theorem 2.10 is that with S = M0[G; I,;P ] wehave

rank(S) rank(G) + max(|I|, ||).

We obtain this bound by joining together a generating set for T = S\ and a gener-ating set for G. In each case, exactly where the answer lies between max(|I|, ||)and rank(G) + max(|I|, ||) will depend on the contribution that is made bythe idempotents of S.

2.4 Finite 0-simple semigroups and their associated

graphs

Given an element (i, g, ) S = M0[G; I,;P ] we may visualise this triple astwo vertices i and joined by a directed edge labelled with g:

(i, g, ) i g // .


Taking this idea further, we may wish to view composition of elements of S ascomposition of such paths so that:

(i, g, ) (j, h, ) i g // j h //

and since (provided pj 6= 0) we have:

(i, g, )(j, h, ) = (i, gpjh, )

we amend our diagram to give:

ig //

pj // jh // .

Grouping together elements of I and those of gives:

: pj

===

====

=

I : i

g@@

jh

@@

which is starting to take the form of a directed bipartite graph with edges labelledby elements of G.

Such representations of completely 0-simple semigroups have been exploitedwith success in the past. The first place that such an idea appears in the literatureis in [41]. In this paper Graham uses the graph theoretic approach to describe allmaximal nilpotent subsemigroups ofM0[G; I,;P ] (a semigroup T is nilpotent iffor some n N we have Tn = {0}). Moreover, necessary and sufficient conditionsare given for a completely 0-simple semigroup to have a unique maximal nilpotentsubsemigroup. Secondly, in the same paper a new normal form is introducedfor completely 0-simple semigroups (see Section 2.10 for more details on this).This normal form is used to give a general description of the form that themaximal subsemigroups of an arbitrary finite completely 0-simple semigroup musttake. This local result was later successfully used by Graham, Graham andRhodes in [40] to give the form of the maximal subsemigroups of arbitrary finitesemigroups. In [56] the bipartite graph representation was used to describe thesubsemigroup generated by the idempotents of a completely 0-simple semigroupS and in [54] Houghton considered the homological properties of these graphs.

Here we will define three graphs, each with a different purpose. The firsthelps us find what a given subset of S generates. The second facilitates the study

Section 2.4 27

of E(S) and the third gives us the concept of connectedness in S.

Graph 1: (S : A)

Given S =M0[G; I,;P ] and A S where 0 / A we define a bipartite digraph(S : A) with labelled edges in the following way. The vertex set of (S : A) isI , where I and are assumed to be disjoint. Edges from I to representelements of A and edges from to I represent idempotents of S in the followingway:

(i) corresponding to each a = (i, g, ) A there is an edge i g labelled withg;

(ii) corresponding to each non-zero entry pj P there is an edge pj jlabelled with pj .

Note that the graph (S : A) is allowed to have multiple edges, so whendescribing a path in this graph it is not enough to just give an ordered list ofvertices that the path is to traverse.

Definition 2.16. Let f = ig be an edge from I to and e = pj j be an

edge from to I in (S : A). We define the functions V and W :

V (f) = g, V (e) = pj , W (f) = (i, g, ) A.

Let x, y I and let p = (e1, e2, . . . , ek) be a directed path in (S : A) startingin x and ending in y. Then we write

V (p) = V (e1)V (e2) . . . V (ek) G,

and call this the value of the path p. We will use Px,y with x, y I todenote the set of all paths starting at x and ending at y in (S : A) and defineVx,y = {V (p) : p Px,y}. We call paths that start in I and end in the validpaths. For every valid path p = (f1, e1, f2, e2, . . . , fk1, ek1, fk) we write

W (p) =W (f1)W (f2)W (f3) . . .W (fk) S.

Clearly if p is a valid path from i to then

W (p) = (i, V (p), ).


There is a clear correspondence between non-zero products of elements of Aand valid paths in the graph (S : A). As a consequence of this correspondencewe have the following straightforward lemma:

Lemma 2.17. Let S = M0[G; I,;P ] and let A S. If R is the set of validpaths in (S : A) then

A = {W (p) : p R} {0}.

Proof. This is obvious from the definitions.

Graph 2: (P )

When A = E(S) the graph (S : A) = (S : E(S)) takes a particularly niceform. Since E(S) = {(i, pi1, ) : pi 6= 0} {0} every edge f = i pi has a corresponding reverse edge e(f) =

pi1

i. In this situation we cansimplify the graph (S : E(S)) in the following way. Let (P ) denote theunderlying undirected graph of (S : E(S)) noting that (P ) has precisely oneedge corresponding to each non-zero pi of P . In (P ) the edges are unlabelledbut we will still assign values to the paths through the graph. The value of thepath pi = z1 z2 . . . zt is defined to be

V (pi) = (z1, z2)(z2, z3) . . . (zt1, zt)

where(i, ) = p1i , (, i) = pi, i I, ,

and Px,y and Vx,y have the same meaning as before. Note that the graph (P )does not have multiple edges and so paths in the graph are uniquely determinedby ordered lists of vertices. Given two vertices x and y in (P ) we write x 1 yif there is a path from x to y in the graph (P ). This connectedness relation 1on the graph (P ) is an equivalence relation on the set I and in [56] Howieproves the following result:

Theorem 2.18. [56, Theorem 1] Let S = M0[G; I,;P ] be a completely 0-simple semigroup, let E be the set of idempotents in S. Then

E = {(i, a, ) S : i 1 and a Vi,} {0}.

Section 2.4 29

Graph 3: (HS)

Given C I we will let (C) denote the undirected graph with set of verticesC and two vertices (i, ) and (j, ) adjacent if and only if i = j or = . Inparticular given S =M0[G; I,;P ] we define HS I as the set of coordinatesof the group H-classes of S, that is

HS = {(i, ) I : Hi is a group} = {(i, ) I : pi 6= 0}.

We will show, in Lemma 2.24, that the graph (HS) is connected if and only ifthe graph (P ) is connected. We say that S is connected if and only if (HS)(or equivalently (P )) is connected (see Figure 2.1 for examples).

Also, for I I and we say that I is a connected componentof S precisely when the subgraph of (HS) induced by the vertices I is aconnected component of (HS).

Example 2.19. Let G = S3 = {(), (1 2), (1 3), (2 3), (1 2 3), (1 3 2)}, the sym-metric group of degree three. Define S =M0[G; {i1, i2, i3}, {1, 2, 3};P ] where

P =

() (1 2) 00 0 (1 3 2)(2 3) 0 ()

.Let

A = {(1, (1 2), 1), (2, (), 2), (3, (1 2 3), 3)} S.Then, for this example, the three graphs defined above are given below.

(1,1) (1,3)

(2,1)

(3,2)

(3,3)

1

AAAA

AAAA

2

AAAA

AAAA

3

nnnnnn

nnnnnn

nnn

i1 i2 i3

1

()

(12)

AAA

AAAA

A 2(132)

AAA

AAAA

A 3

()

EDBC

GFBC (23)iiRRRRRRRRRRRRRRRRRR

i1

(12)

II

)

i2

()

UU

)

i3

(123)

UU

)

(HS) (P ) (S : A)

Since (HS) is connected, the semigroup S is a connected completely 0-simplesemigroup.

Since F (S) = E(S) is a subsemigroup of S and Hi is a subgroup of S theintersection F (S) Hi is a subsemigroup of Hi in S and therefore must be a


Figure 2.1: Two egg-box pictures of D-classes of completely 0-simple semigroups.The shaded boxes are the group H-classes.

Connected Disconnected

subgroup of S. But what does this group look like? Clearly

F (S) Hi {(i, k, ) : k K}

where K is the subgroup of G generated by the non-zero entries of the matrix P .In general, however, these two sets are not going to be equal.

Example 2.20. Let G be the cyclic group of order 5 written multiplicativelyand generated by a: so G = {a0, a1, a2, a3, a4}. Let S = M0[G; {1, 2}, {1, 2};P ]with:

P =

(1 a0 1

).

From the discussion above we deduce that:

F (S) H11 1G 1 = H11.

However, by Theorem 2.54, the semigroup S is isomorphic to

T =M0[G; {1, 2}, {1, 2};Q]

where:

Q =

(1 10 1

)and hence for every group H-class Hi of T = S we have:

F (T ) Hi = {(1, a0, 1)}.

Section 2.4 31

Since T = S it follows that:

F (S) H11 = {(1, a0, 1)} 6= 1G 1.

This example is important for the following reason. If we are interested indetermining E(S) for S a finite 0-simple semigroup then, in the example above,the latter of the two representations is a more useful one. It satisfies the propertythat the subsemigroup generated by the idempotents intersected with a groupH-class is isomorphic to the subgroup of G generated by the non-zero entries inthe matrix P . In fact, such a nice normalization always exists. This is calledGraham normal form (see [41]) and will be discussed in detail in Section 2.10.

Returning to the problem of describing the group F (S)Hi we now show howthis group relates to a group of paths in the graph (P ). Let S =M0[G; I,;P ]be connected and let (1I , 1) I with p11I 6= 0 so that H1I1 is a groupH-class of S.

Lemma 2.21. [81, Lemma 4.3] The mapping : H1I1 G defined by

((1I , g, 1)) = gp11I

is a group isomorphism. It maps H1I1 F (S) onto V1I1p11I .

Proof. It is routine to check that the map is an isomorphism. The second asser-tion follows from Theorem 2.18.

We can actually say a lot more about the subgroups Vipi of G.

Lemma 2.22. Let Hi and Hj be group H-classes of a connected completely0-simple semigroup S = M0[G; I,;P ]. Then the subgroups Vipi and Vjpjare conjugate in G.

Proof. Since S is connected we can fix a path pi in (S : E(S)) from to i. Weclaim that with g = p1j V (pi) we have

gVipig1 = Vjpj .

It is sufficient to show:

p1j V (pi)VipiV (pi)1 Vj.


Let p Pi. In the graph (S : E(S)) this path may be extended to

jp1j

pii

p

pi i pi1

Pj.

It follows thatp1j V (pi)V (p)piV (pi)

1 Vj.

Since this is true for every p Pi the result follows.

As promised earlier we now describe all the bases of the aperiodic Brandtsemigroup Bn. The structure matrix P of Bn is the n n identity matrix In.Therefore the graph (P ) has the following form.

1 2 . . . n

i1

i2. . .

in.

Let A Bn. Let G(Bn, A) be the graph given by contracting the edges (k, ik),for all 1 k n, in the graph (Bn : A). Thus the graph G(Bn, A) is isomorphicto the graph with vertex set {1, . . . , n} and set of directed edges equal to A. Itfollows from Lemma 2.17 that A generates Bn if and only if the graph G(Bn, A) isstrongly connected (i.e. there is a directed path between every pair of vertices).It follows from [8, Corollary 7.2.3] that if |A| = n then G(Bn, A) is stronglyconnected if and only if it is isomorphic to the following graph.

1 //2?

????

???

n?? 3

7

OO

4

6

__???????? 5

oo

This proves the following result.

Proposition 2.23. Let S = Bn be the nn aperiodic Brandt semigroup and let Sn. Then

A = {(i, i) : 1 i n}

generates S if and only if is a n-cycle in Sn.

Section 2.5 33

pi

pi

tti ii i

i ii i

i ii i

i

i

p

OO

pi1

44iiiiiiiiiiiij

p1j

OO

Figure 2.2: Diagram showing that the groups Vjpj and Vipi are conjugate inG.

2.5 Connected completely 0-simple semigroups

The question of finding a formula for rank(S) divides into two cases: the casewhen S is connected and the case when it is not. The connected case was dealtwith in [81] and in what follows we will extend these results to deal with the dis-connected case. For what remains of this section S will denote a finite connectedcompletely 0-simple semigroup. We now give the details of a number of resultsfrom [81] which will be needed later on.

Lemma 2.24. [81, Theorem 2.1] The following conditions are equivalent forany completely 0-simple semigroup S:

(i) (HS) is a connected graph;

(ii) (P ) is a connected graph;

(iii) F (S) Hi 6= for any i I and any .

Proof. ((i) (ii)) The graph (HS) is disconnected if and only if there existvertices (i, ) and (j, ) in different connected components of (HS) which is trueif and only if the edges {i, } and {j, } are in different connected components ofthe graph (P ). Such a pair of edges exists if and only if (P ) is not connected.((ii) (iii)) Since the graph (HS) is connected it follows that the graph (P )is connected. Given some i I and let p = (e1, e2, . . . , ek) be a directedpath in (P ) starting at i and ending at . This is a valid path, as defined inthe previous section, and the corresponding element W (p) F (S) belongs to theH-class Hi. Since i and were arbitrary it follows that F (S) Hi 6= for all(i, ) I . ((iii) (i)) Suppose that Hi F (S) is empty. Then it followsfrom the definition of the graph that there is no path from i to in the graph(P ). It follows that (P ) is not connected and so neither is (HS).

In the next lemma we introduce the function (i, , j, ). This function willplay a crucial role in what follows. The family of functions (i, , j, ) where


i, j I and , allow us to use the idempotents of connected completely0-simple semigroups to move between the H-classes. More than this, whenmoving between two group H-classes we are able to map in an isomorphic way.Lemma 2.25. [81, Theorem 2.1] Let S = M0[G; I,;P ] be a connected com-pletely 0-simple semigroup. For any i, j I and any , there existp(i, , j, ), q(i, , j, ) F (S) such that the mapping (i, , j, ) : Hi Hjdefined by

(i, , j, )(x) = p(i, , j, )xq(i, , j, )

is a bijection.The elements p(i, , j, ), q(i, , j, ) can be chosen so that

(i, , j, )1 = (j, , i, )

and (i, , j, ) is a group isomorphism if both Hi and Hj are groups.

Proof. First we show that a bijection can be found with the given properties.Then we show that in the special case where Hi and Hj are both groups thisbijection may be chosen to be an isomorphism.

In the graph (P ) choose and fix a path pi which starts at j and finishes ati. Such a path exists since the graph (P ) is connected. Say this path is:

j 1 j2 2 . . . jm m i.

This is not a valid path but if we shorten it slightly we do get a valid path. Let pi

denote the path pi but restricted from j to m. In a similar way let be a fixedpath in the graph (P ) connecting to . Such a path exists since the graph isconnected. Say the path is:

i1 1 i2 . . . n in .

Once again, this path is not valid in (P ) but if we shorten it, by removing theinitial vertex, we get a path starting at i1 and ending at which is valid. Callthis path . By the definition of (P ) we haveW (pi) F (S) andW () F (S).

Now define p(i, , j, ) =W (pi) and q(i, , j, ) =W (). The map

(i, , j, )(x) = p(i, , j, )xq(i, , j, )

is a bijection from Hi to Hj since it sends (i, g, ) Hi to (j, V (pi)gV (), )where V (pi), V () G are both fixed.

Section 2.5 35

Now define the path pi1 from i to j to be the reverse of the path pi andthe path 1 to be the reverse of the path . The paths pi1 and 1 aredefined in the analogous way to above and then define p(j, , i, ) = W (pi1)and q(j, , i, ) =W (1). In this way the map (j, , i, ) maps (j, h, ) Hjto (i, V (pi)1hV ()1, ). Therefore for every (i, g, ) Hi we have:

(j, , i, )((i, , j, )((i, g, ))) = (j, , i, )((j, V (pi)gV (), ))

= (i, V (pi)1V (pi)gV ()V ()1, )

= (i, g, )

and so (i, , j, )1 = (j, , i, ).Now consider the special case where both Hi and Hj are group H-classes.

This means that the edges ip1i and j p

1j belong to the graph (P ). This

gives some control over the choice of the paths pi and in the above construction.Carry out the same process as above but this time once pi has been fixed we maydefine

= pi i pi1 j p

1j .

Now the map = (i, , j, ) maps (i, g, ) Hi to (j, V (pi)gpiV (pi)1p1j , ) Hj. Again, this map is clearly a bijection. Moreover, it is an isomorphism since:

((i, g, )(i, h, )) = (i, gpih, )

= (j, V (pi)(gpih)piV (pi)1p1j , )

= (j, V (pi)(gpi)(hpi)V (pi)1p1j , )

= (j, V (pi)(gpi)V (pi)1p1j pjV (pi)(hpi)V (pi)1p1j , )

= (j, V (pi)gpiV (pi)1p1j , )(j, V (pi)hpiV (pi)1p1j , )

= ((i, g, ))((i, h, )).

Lemma 2.26. Let Hi and Hj be group H-classes of the connected completely0-simple semigroup S. There is an isomorphism : Hi Hj such that (F (S)Hi) = F (S) Hj.

Proof. The map = (i, , j, ) : Hi Hj by Lemma 2.25 is an isomorphism.We claim that (F (S) Hi) = F (S) Hj. Indeed, if x F (S) Hi then(x) F (S)xF (S) F (S)3 = F (S) and so (F (S) Hi) F (S) Hj. Since is a bijection and since, by Lemma 2.22 the groups F (S)Hi and F (S)Hj


are isomorphic, it follows that (F (S) Hi) = F (S) Hj.

So the subgroups F (S) Hi and F (S) Hj are not only isomorphic butthey sit inside their respective group H-classes in the same way. The followingresult now follows.

Corollary 2.27. Let S be a connected completely 0-simple semigroup with Hiand Hj two group H-classes of S. Then

rank(Hi : F (S) Hi) = rank(Hj : F (S) Hj).

The next lemma, roughly speaking, shows us how effective a given subset Aof S can be in helping to generate the elements of a fixed group H-class Hi.

Lemma 2.28. [81, Lemma 3.4] Let A = {a1, . . . , ar} S where aj Hijj , j =1, . . . , r and let Hi be a group H-class. If we write

B = {(i1, 1, i, )(a1), . . . , (ir, r, i, )(ar)} Hi

thenF (S) A Hi = (F (S) Hi) B.

Proof. () First note that by definition B F (S)AF (S) and so B = B Hi F (S)AF (S) Hi. It follows that:

(F (S) Hi) B (F (S) Hi) (F (S)AF (S) Hi) F (S) F (S)AF (S) Hi F (S) A Hi.

() First observe that A F (S)BF (S) since for a A Hj we have

a = (i, , j, )((j, , i, )(a)) = p(i, , j, )(j, , i, )(a)q(i, , j, )

F (S)(j, , i, )(a)F (S) F (S)BF (S).

It follows that:

F (S) A Hi F (S) F (S)BF (S) Hi F (S) B Hi.

Also:F (S) B Hi = E(S) B Hi (F (S) Hi) B.

Section 2.5 37

Indeed, any product in E(S) B Hi has the form:

e1 . . . er1b1er1+1 . . . er2b2er2+1 . . . erk1bk1erk1+1 . . . erk

where each bm Hi and en E(S) for all n. Let ei denote the idempotent ofHi which is the identity of this group H-class. The above product is equal to:

(e1 . . . er1ei)b1(eier1+1 . . . er2ei)b2(eier2+1 . . . erk1ei)bk1(eierk1+1 . . . erk)

which has the formf1b1f2b2 . . . bk1fk

where fl F (S) Hi and bl B for all l. The reason that f1 and fk belong toHi is because the entire product f1b1f2b2 . . . bk1fk belongs to Hi. It followsthat:

F (S) A Hi F (S) B Hi (F (S) Hi) B

as required.

Using the previous result we may obtain a result concerning the relative rankof any subset of F (S) in S.

Lemma 2.29. Let S be a connected completely 0-simple semigroup with Hi agroup H-class and U F (S). Then

rank(S : U) rank(Hi : Hi F (S)).

Proof. Let V S be such that U V = S, say V = {v1, . . . , vm} S wherevk Hikk (k = 1, . . . ,m). If we write

B = {(i1, 1, i, )(v1), . . . , (im, m, i, )(vm)} Hi

then by Lemma 2.28 we have

F (S) V Hi = (F (S) Hi) B.

Now since U V is a generating set for S, we have

Hi = S Hi = U V Hi = F (S) U V Hi= F (S) V Hi = (F (S) Hi) B.


Therefore|V | |B| rank(Hi : F (S) Hi).

This leads to the following corollary which gives an important lower boundfor ranks of connected completely 0-simple semigroups.

Corollary 2.30. [81, Lemma 3.6] If Hi is a group H-class then

rank(S) rank(Hi : Hi F (S)).

Proof. By Lemma 2.29 we have:

rank(S) = rank(S : ) rank(Hi : Hi F (S))

as required.

2.6 Arbitrary completely 0-simple semigroups

Now we look at the case when (HS) is not necessarily connected. For theremainder of this chapter, unless otherwise stated, S will denote a completely 0-simple semigroup, represented as a Rees matrix semigroupM0[G; I,;P ], with kconnected components I11, . . . , Ikk so the matrix P has the form suggestedby the following picture:

I1 I2 . . . Ik

12...k

C1 0

C2. . .

0 Ck

.

We will first consider some properties that all generating sets of S must have.

Definition 2.31. We say that A S is an H-class transversal generating set if

A Hi 6= for all i I and .

For example, Theorem 2.24 tells us that (HS) is connected if and only if E(S) isanH-class transversal generating set. Clearly the smallest size such a set can haveis max(|I|, ||). In fact, as a consequence of the results of Section 2.3, concerning

Section 2.6 39

the rank of an arbitrary rectangular 0-band, we know that we can always find atleast one H-class transversal generating set with this minimal size.

Lemma 2.32. There exists A S such that |A| = max(|I|, ||) and A is anH-class transversal generating set.

Proof. The assertion follows from Theorem 2.10 and from the fact that themapping (i, g, ) 7 (i, ) defines an epimorphism from S onto a rectangular0-band.

An H-class transversal generating set with size max(|I|, ||) will be the firstbuilding block we will use when constructing minimal generating sets for S. Wewill call an H-class transversal generating set with size max(|I|, ||) an H-classtransversal basis.

Definition 2.33. We call C I component connecting coordinates if (HSC) is connected. Similarly we will call A S a component connecting set if{(i, )|(i, g, ) A} I is a set of component connecting coordinates.

If (HS) has k connected components then the smallest size a component con-necting set can have is k 1. We will call these the minimal connecting sets.

Lemma 2.34. Every component connecting set D has a subset E D that isminimal.

Proof. Let A be the coordinates of the component connecting set D. LetC1, . . . , Ck be the connected components of (HS). Construct a new graph

with vertices C1, . . . , Ck and Ci adjacent to Cj if and only if there is some ci Ci,cj Cj and a A such that {ci, a} and {a, cj} are edges in (HS). Then isa connected graph with k vertices and so has a spanning tree with k 1 edges.This spanning tree corresponds, in an obvious way, to a minimal set of coordinatesA A which in turn correspond to a subset E D that is minimal.

Lemma 2.35. Every H-class transversal generating set is component connecting.

Proof. Suppose otherwise. Let A be an H-class transversal generating set thatis not component connecting. Then we can choose (i, ), (j, ) I such that(i, ) and (j, ) are in different connected components of (HS A). It followsthat i and are in different connected components of the graph (S : A) and soby Lemma 2.17 we have A Hi = , a contradiction.


Corollary 2.36. Every H-class transversal generating set (and in particular ev-ery H-class transversal basis) has a component connecting subset which is mini-mal.

Example 2.37. Let G = {1, a} be the cyclic group of order 2 and let

S =M0[G; {1, . . . , 4}, {1, . . . , 4};P ]

where

P =

a 0 0 0a 1 0 00 0 a 00 0 0 1

S =

.

(1,1) (1,2) (3,3) (4,4)(2,2)

(HS)

The graph (P ) has 3 connected components.

(i) C = {(1, 3), (2, 1), (3, 4), (4, 2)} I is a set of component connectingcoordinates.

(ii) A = {(1, a, 3), (2, 1, 1), (3, 1, 4), (4, a, 2)} S is an H-class transversal basiswhich is a component connecting set.

(iii) T = {(1, a, 3), (3, 1, 4)} A is a minimal component connecting subset ofA.

We now define rmin which is the most complicated term that will appear inthe formula for the rank of a 0-Rees matrix semigroup. We use Map(X,Y ) todenote the set of all maps from a set X into a set Y .

Definition 2.38. Given the structure matrix P of S we define PC, where C I and Map(C,G) to be a new || |I| matrix with entries pi where

pi =

{(, i) if (, i) Cpi otherwise.

Also let SC, = M0[G; I,;PC,]. When |C| = 1 we will use the more relaxednotation S(, i, (, i)).

Section 2.6 41

Since, depending on the circumstances, we may need to view sets of coordi-nates sometimes as subsets of I and at other times as subsets of I wemake the following definition.

Definition 2.39. Given A I we define AT as

AT = {(, i) : (i, ) A} I,

and call this set the transpose of A.

Definition 2.40. Let Hi be any group H-class of S and let C I be a setof component connecting coordinates with minimal size. Then define

rmin = minMap(CT ,G)

(rank(Hi : Hi F (SCT ,))).

If S is connected then C = and rmin = rank(Hi : Hi F (S)). OtherwiseS is not connected and we associate the family {SCT , : Map(CT , G)} ofconnected completely 0-simple semigroups with S. We then look through thisfamily searching for a member that minimizes the relative rank of F (S) in agroup H-class.

We have to show rmin is well defined i.e. that it does not depend on the choiceof Hi or on the choice of C. First we note that, since SCT , is connected, byCorollary 2.27 the number rmin is independent of the choice of Hi. That rmindoes not depend on the choice of C will eventually be proved in Lemma 2.43.

Lemma 2.41. Suppose that P has two connected components 1I1 and 2I2,and let Hi be a group H-class. If (, j), (, k) (1 I2) (2 I1) then forevery g G there exists g G such that

Hi F (S(, j, g)) = Hi F (S(, k, g)).

Proof. There are essentially two cases to consider.

Case 1: (, j), (, k) 1I2. Let (i, ) I be arbitrary. We will show thatgiven (, j), (, k) and g G we can choose g G so that if we let S1 = S(, j, g)with structure matrix P1 and S2 = S(, k, g) with matrix P2, then we have

V S1i = VS2i .

Here V Ui denotes the set of values of all paths from i to (so far denoted simplyas Vi) in the graph (Q) where Q is the structure matrix of U . This convention


will be used throughout and will also apply to sets of paths Pi. Let us chooseand fix a path pi in PS and define w = V (pi) V S. Also, choose and fix apath pijk in PSjk defining wjk = V (pijk) V Sjk. Such paths exist since I1 1 andI2 2 are both connected components in (P ). Now define

g = wgwjk V SgV Sjk

so thatg = w1 g

w1jk V SgV Skj .

We observe that (P1) is connected and is precisely (P ) with the extra edge

g j. Also (P2) is connected and is precisely (P ) with the extra edge

g k. Let p be an arbitrary path from i to in (P1). We show that there isa corresponding path p in (P2) with the same value. While following the pathp whenever we come across

. . . j . . .

we replace it with. . . k j . . . .

Recall that the graph (P ) is directed and has unlabelled edges. Also, paths areuniquely determined by giving a list of vertices. Above we move from to usingthe fixed path pi and from k to j using the path pikj . Clearly V (p) = V (p) bydefinition of g and so V S1i V S2i . Similarly V S2i V S1i and the result followsfrom Theorem 2.18. The case where (, j), (, k) 2 I1 is dual to this case.

Case 2: (, j) 1 I2 and (, k) 2 I1. We follow a very similar argumentbut this time let wj V Sj and wk V Sk, which is possible since I1 1 andI2 2 are both connected components in (P ). Then g is chosen so that thepaths

k

and j k

have the same value.

Lemma 2.42. Let T be the family of all spanning trees of the complete graph

Section 2.7 43

Kn. If denotes the graph with vertex set T and edges defined by

(T1, T2) E() e1, e2 E(Kn) : (T1 {e1}) {e2} = T2

then is connected.

Proof. Observe that the set of acyclic subsets of the edge set of a graph definesa matroid (see [15, Chapter 12]). Then since in particular the spanning trees ofKn are acyclic edge sets they are independent sets in the corresponding matroid.That there is a path between the vertices T1, T2 T is now a consequence of theexchange axiom for matroids.

Combining the previous two lemmas, we conclude the following which tells usthat rmin is indeed well defined.

Lemma 2.43. Let S = M0[G; I,;P ] be a completely 0-simple semigroup withconnected components I1 1, . . . , Ik k. Let C,D I be two sets ofminimal component connecting coordinates. If Hi is a group H-class in S then

minMap(CT ,G)

(rank(Hi : Hi F (SCT ,))) = minMap(DT ,G)

(rank(Hi : Hi F (SDT ,))).

Proof. First we note that (HS C) and (HS D) are connected. We removeone c C, changing (HS C) into (HS (C \ {c})) which has two connectedcomponents. Now by Lemma 2.42 we can find d D such that (HS(C \{c}){d}) is connected and by Lemma 2.41 we can replace by so that rank(Hi :Hi F (SCT ,

ThesisFinal a Graph Theoretic Approach to Combinatorial

Documents