CHAPTER 1

Introduction

The concept of a fundamental semigroup, which is a semigroup that cannot be shrunk homomorphically without collapsing idempotents together, was introduced by Munn [25] in 1966, who developed an elegant theory within the class of inverse semigroups, which inspired many researchers in subsequent decades. There is a natural partial order on the set of idempotents of any semigroup, which is given by e ≤ f ⇐⇒ ef = fe = e. In any inverse semigroup idempotents commute, so the product of any two idempotents is idempotent. The set of idempotents becomes a semilattice under ≤, where the greatest lower bound of two idempotents e and f is the product ef. Munn constructed a semigroup

TE from an arbitrary semilattice E, which he proved is the maximum fundamental inverse semigroup with semilattice of idempotents E. Any fundamental inverse semigroup with

semilattice of idempotents E must be isomorphic to a full subsemigroup of TE. Also, for any inverse semigroup with semilattice of idempotents E, Munn constructed a fundamental representation

φ : S → TE with kernel µ = {(a, b) ∈ S × S | (∀e ∈ E) a−1ea = b−1eb} which is the maximum idempotent separating congruence on S. These results demonstrate

the importance of TE, since every inverse semigroup with semilattice of idempotents E is a coextension of a fundamental inverse semigroup S/µ with the same semilattice of idem-

potents, which is a subsemigroup of TE. For any set X the symmetric inverse semigroup

IX = {partial one-one mappings : X → X}

is a fundamental inverse semigroup, which is an example of Munn’s construction TE. For more details about Munn’s theory of fundamental inverse semigroups see [22] Section 5.4.

Several authors have generalised Munn’s work to the class of fundamental regular semi-

groups. Each author constructs a semigroup TE from an algebraic structure E related to the set of idempotents of a regular semigroup, which is the maximum fundamental regular semigroup with respect to the structure E. Hall noted in his review of [27] that since each 1 1. INTRODUCTION 2

construction is the maximum fundamental regular semigroup with a given partial groupoid of idempotents, they must be isomorphic.

In 1973 Hall [20] used the semigroup generated by the set of idempotents of a regular semigroup, which is guaranteed to be regular by a result of FitzGerald [15]. He also used a generalisation of Munn’s fundamental representation of an inverse semigroup to a fundamental representation ∗ φ : S → TS/L × TS/R of a regular semigroup S, with kernel

µ = {(a, b) ∈ S × S | for some inverses a′ of a and b′ of b, aa′ = bb′, a′a = b′b and a′ea = b′eb for each idempotent e ≤ aa′} which is the maximum congruence contained in H , and also the maximum idempotent separating congruence on S.

In 1974 Grillet [18, 19] used the posets I = S/R and Λ = S/L where S is a regular semigroup, together with a cross-connection which is a pair (Γ, ∆) of order preserving mappings Γ : Λ → I∗ and ∆ : I → Λ∗ with a certain property. For more details on Grillet’s construction see [17] Section VIII.1 and Section VIII.2.

The natural partial order ≤ provides a coarse means of comparing idempotents. In 1974- 1975 Nambooripad [26, 28] refined this to a pair of intertwining preorders and (following the notation of Easdown [3, 5]) which are defined by e f ⇐⇒ fe = e, e f ⇐⇒ ef = e. The arrows and satisfy the intertwined antisymmetric property:

e f e ⇒ e = f.

In an arbitrary semigroup there may be some noncommuting idempotents, so the product of two idempotents may not be idempotent. The set of idempotents is a poset under the intersection = ∩ which is the natural partial order ≤. Nambooripad introduced the concept of a boset, which is a set equipped with the intertwining preorders and , and a partial multiplication such that certain axioms hold. The term boset was coined by Jordan [23], and serves both as an abbreviation of biordered set (due to Nambooripad), and as a generalisation of poset, itself an abbreviation of partially ordered set. The set of idempotents of a regular semigroup is a regular boset, which is a boset in which all sandwich sets are nonempty (see [26, 28] and also Pastijn [29]). The sandwich set of an ordered pair of boset elements is a generalisation of the greatest lower bound of a 1. INTRODUCTION 3

pair of poset elements. Nambooripad’s construction used regular bosets. For more details on Nambooripad’s construction see [28].

In 1975-1976 Clifford [1] used the concept of a warp, which is a special partial groupoid, and regarded the set of idempotents of a regular semigroup as a partial groupoid rather than a boset.

In 1978 Nambooripad [27] explored the relationship between his construction based on bosets, and Grillet’s construction based on posets and cross-connections.

In 1984 Easdown and Hall [3] generalised the fundamental representation of a regular semigroup further to a fundamental representation

∗ φ : S → TX∪{∞} × TY ∪{∞} of an arbitrary semigroup S, where X and Y are the sets of regular L -classes and R-classes of S respectively. The kernel of φ is the congruence

µ = {(a, b) ∈ S × S (∀x ∈ E) each of x R xa, x R xb implies xa H xb and each of x L ax, x L bx implies ax H bx}.

which is the maximum H -congruence on S (the notion of an H -congruence is technical and explained later in this thesis). A theory of fundamental semigroups was developed by Easdown [9, 3], Edwards [10] and Hall [20, 3]. Easdown proved that an arbitrary semigroup is a coextension of a fundamental semigroup in which the relationships between the idempotents of the original semigroup have been preserved.

Easdown and Hall also introduced a representation, which we will also call φ here, of an arbitrary boset E by transformations and dual transformations. The idempotent-generated

semigroup hEφi is fundamental and sits inside a construction TE introduced by Jordan [23] in 2002 which generalises the Hall-Grillet-Nambooripad-Clifford construction of the regular case. Jordan gave another proof that TE is the maximum fundamental regular semigroup with boset of idempotents E whenever E is a regular boset, reproducing the earlier results.

We finish this introduction with two contrasting examples, discussed in more detail later. If G is any group, a semigroup G ⊔ G where G = {g | g ∈ G} may be defined with 1. INTRODUCTION 4

1 = G

··· ← elements of G

Figure 1.1. A semigroup with a regular boset of idempotents multiplication g · h = gh, g · h = gh, g · h = g · h = h,

for each g, h ∈ G. The semigroup G ⊔ G in Figure 1.1 turns out to be fundamental and is

an example of a subsemigroup of TE where E is a regular boset. The full group of units of

TE in this example would be the symmetric group on the set G, in which G is embedded under the usual right regular representation.

It is the main aim of this thesis to generalise the result that TE is maximum fundamental, beyond the class of regular semigroups. We prove that TE is fundamental for any boset E. We will see in Section 4.6 that when E is a nonregular boset, the boset of idempotents of TE might be larger than E, so there is no hope of TE being the maximum fundamental semigroup with boset of idempotents E in general. However, we prove that every funda- mental semigroup which is generated by regular elements and has boset of idempotents E is isomorphic to a subsemigroup of TE with boset of idempotents Eφ, and conversely every subsemigroup of TE which is generated by regular elements and has boset of idempotents Eφ is fundamental. When E is a regular boset this reproduces the result of Nambooripad’s that TE is the maximum fundamental regular semigroup with boset of idempotents E.

When E is a regular boset, TE is also regular, but when E is a nonregular boset, TE may be regular or nonregular. We investigate the regularity of TE for some nonregular bosets. A class of bosets, called sawtooth bosets which are named for the shape of the boset diagrams, is introduced which contains infinitely many regular and nonregular examples. The fundamental semigroup described pictorially in Figure 1.2 contains five idempotents e,f,g,h and k, and one nonregular element fe, and is an example of a subsemigroup of TE, OVERVIEW OF THESIS 5 e f

fe

h g k k k ke hf k ef

Figure 1.2. A semigroup with a nonregular boset of idempotents

in fact an idempotent-generated subsemigroup, where E is a nonregular boset. Its boset is one of the first examples of sawtooth bosets, which are due to Easdown [8]. This boset and its dual are the smallest examples of nonregular bosets of finite semigroups [6]. The regularity of TE where E is a sawtooth boset with 2 teeth has been classified by Jordan [23]. We investigate his work, and also define a subclass of sawtooth bosets, called cyclic sawtooth bosets, which contain infinitely many regular and nonregular examples, and for which TE is always regular.

Overview of Thesis

We begin in Chapter 2 by recalling some well-known facts about semigroup theory. We also reintroduce the concept of a fundamental semigroup, and discuss further Munn’s theory of fundamental inverse semigroups.

In Chapter 3 we introduce Nambooripad’s concept of a boset, by looking at some examples involving linear operators of vector spaces, endomorphisms of groups, and idempotents of semigroups. We then define bosets abstractly and introduce the class of sawtooth bosets.

We begin Chapter 4 with the theory of the fundamental representation φ, due to Easdown,

Edwards and Hall. We then define Jordan’s construction TE for an arbitrary boset E, and describe its relationship to the fundamental representation φ. We finish the chapter with some useful results about products in TE, and by investigating TE for various classes of bosets: regular bosets, posets and semilattices.

In Chapter 5 we prove two main results of this thesis. First that TE is fundamental for any boset E, and second, that fundamental semigroups which are generated by regular OVERVIEW OF THESIS 6

elements and have boset of idempotents E are precisely subsemigroups of TE which are generated by regular elements and have boset Eφ. We also provide an alternative proof

that an idempotent generated subsemigroup SE of TE is fundamental for any boset E.

In Chapter 6 we investigate the regularity of TE and SE for some sawtooth bosets. We prove that TE and SE are both regular for any cyclic sawtooth boset, and discuss Jordan’s regularity criteria for TE and SE where E is a sawtooth boset with 2 teeth. CHAPTER 2

Semigroups

In this chapter we recall some well-known terminology and simple facts, relating them wherever possible to the ideas introduced later in the thesis. The concept of a fundamental semigroup is carefully explained, as well as Munn’s [25] theory of fundamental inverse semigroups.

2.1. Preliminaries

A set S is a partial groupoid with respect to a partial binary operation · if for some pairs of elements a, b ∈ S there is an element a · b ∈ S which is the product of a by b. The set

DS = {(a, b) | a · b is defined} is called the domain of S. In practice, the multiplication is often denoted by juxtaposition.

A partial groupoid S is a groupoid if the partial binary operation on S is full, that is, if

DS = S × S. A pair of elements a, b ∈ S commute if ab = ba, and a subset X of S is commutative if each pair of elements in X commute.

A groupoid S is a semigroup if the binary operation on S is associative, that is, if a(bc)=(ab)c for all a, b, c ∈ S. For any semigroup S, the dual semigroup of S is the semigroup obtained from S by reversing all of the products, which is the semigroup S∗ = {a∗ | a ∈ S}, whose underlying set is chosen to be in a one-one correspondence with S under the mapping a 7→ a∗, with respect to the operation a∗b∗ =(ba)∗. Observe that for any statement that is true for all semigroups, the dual statement, that is, the statement obtained by reversing the products in the original statement, is also true for all semigroups. This duality, and variations involving bosets introduced in Chapter 3 and Chapter 4, will be used throughout this thesis. 7 2.1. PRELIMINARIES 8

An element e ∈ S is idempotent if e2 = e. Denote the set of idempotents in S by E(S) or simply by E. There is a natural partial order on E which is given by,

e ≤ f ⇐⇒ ef = fe = e, in which case e is called a zero for f. The order ≤ provides a coarse means of comparing idempotents. In Chapter 3, Nambooripad’s refinement involving two intertwining preorders is introduced. These preorders are denoted by and (following Easdown [3, 5]) and have the property that their intersection ∩ coincides with ≤. A band is a semigroup in which every element is idempotent.

A semilattice is a poset E in which, for each pair of elements e, f ∈ E there is an element e ∧ f ∈ E such that

(i) e ∧ f ≤ e and e ∧ f ≤ f; and (ii) (∀g ∈ E) g ≤ e and g ≤ f implies g ≤ e ∧ f.

The element e ∧ f is called the greatest lower bound of e and f. These two conditions motivate the definition of the sandwich set of a pair of boset elements (Chapter 3). In the case that the boset is a semilattice e ∧ f becomes the unique element in the sandwich set of e and f. An operation may be defined on E by

ef = e ∧ f.

It is easy to check that E becomes a commutative band under this operation. Equivalently, any commutative band E becomes a semilattice under the natural partial order on E. For each pair of elements e, f ∈ E the greatest lower bound of e and f is the product ef.

An element z ∈ S is a right zero (resp. left zero) for a subset X ⊆ S if xz = z (resp. zx = z) for all x ∈ X. A semigroup S is a right zero semigroup (resp. left zero semigroup) if every element of S is a right zero (resp. left zero) for S.

An element z ∈ S is a zero for a subset X ⊆ S if it is both a right zero and left zero for X, that is, if xz = zx = z for all x ∈ X. We also say that z ∈ S is a zero (resp. right zero, left zero) for each x ∈ X if it is a zero (resp. right zero, left zero) for X. If e and f are idempotents then write e f (resp. e f) if e is a right zero (resp. left zero) for f. Arrows can be combined in obvious ways so that e f (resp. e f) means e and f are right zeros (resp. left zeros) for each other, and is the natural partial order ≤ on E. Arrows will be introduced in context in Chapter 3. If there is an element 0 ∈ S which is a zero for S then we say that 2.1. PRELIMINARIES 9

S is a semigroup with zero. If S is any semigroup we can construct a semigroup S0 with zero in the following way. If S has a zero then put S0 = S, otherwise put S0 = S ∪{0} where 0 is a new element and extend the multiplication by

a0=0a =0 for all a ∈ S0.

An element e ∈ S is a right identity (resp. left identity) for a subset X ⊆ S if xe = x (resp. ex = x) for all x ∈ X, and is an identity for X if it is both a right identity and left identity for X, that is, if xe = ex = x for all x ∈ X.

A semigroup S is a monoid if there is an element 1 ∈ S which is an identity for S. If S is any semigroup, a monoid S1 can be constructed from S in the following way. If S has an identity element then put S1 = S, otherwise put S1 = S ∪{1} where 1 is a new element and extend the multiplication by a1=1a = a for all a ∈ S1.

A monoid G is a group if every element g ∈ G has a (group) inverse, which is an element g−1 ∈ G such that gg−1 = g−1g =1. It is well known that group inverses are unique. For each g, h ∈ G the product h−1gh is the conjugate of g by h and is often denoted by gh. If X is any subset of G then Xg is the set of conjugates of elements of X by g. It is very useful in semigroup theory to weaken the notion of inverse, though uniqueness is lost in general.

An element a ∈ S is regular if there is an element a′ ∈ S such that

aa′a = a and a′aa′ = a′.

The element a′ isa(semigroup) inverse for a. Denote the set of inverses for a in S by V (a) and the set of regular elements in S by Reg(S). The condition aba = a for some b ∈ S is sufficient for a to be regular, for then bab is an inverse for a. In this case the elements ab and ba are both idempotent. This idea was extended by FitzGerald [15] to prove that the subsemigroup generated by the set of idempotents of a regular semigroup is regular

(terminology explained below). Let a, b1,...,bn ∈ S such that

ab1 ··· bna = a 2.1. PRELIMINARIES 10

and put

xi = bi+1 ··· bnab1 ··· bi for i =0,...,n where bn+1 ··· bn and b1 ··· b0 represent the empty string. It is easy to check that the elements x0,...,xn are all idempotent. In this thesis they are called FitzGerald idempotents.

A semigroup S is regular if every element in S is regular, and inverse if every element a ∈ S has a unique inverse a−1 ∈ S. Equivalently, a semigroup S is inverse if and only if

(i) S is regular; and (ii) E is commutative.

The proof of this equivalence is not obvious and can be found in [30], Theorem II.1.2, for example. It is well known that if a and b are elements of an inverse semigroup then (ab)−1 = b−1a−1. This follows quickly from the fact that idempotents commute. In regular semigroups which are not inverse there must be some noncommuting idempotents, and one does not expect any nice relationship between inverses of elements and inverses of their product.

Contrasting examples, at two extremes within the class of inverse semigroups, are the following:

(i) Semilattices. Every element is idempotent and equal to its own (unique) inverse. The set of idempotents of any inverse semigroup is a commutative band which becomes a semilattice under the natural partial order. (ii) Groups. Only one element is idempotent, which is the identity element. Every element has a unique (semigroup) inverse which is equal to its group inverse. Any inverse semigroup with exactly one idempotent is a group.

A subset T of a semigroup S is a subsemigroup of S if T is a semigroup under the operation inherited from S, that is, T is closed under multiplication, in which case we may write T ≤ S. If further T contains all of the idempotents in S then T is full.

An equivalence relation σ on a semigroup S is a right congruence (resp. left congruence) if ac σ bc (resp. caσcb) whenever a σ b for all a, b, c ∈ S, and is a congruence if it is both a right congruence and left congruence. Equivalently, σ is a congruence if ac σ bd whenever a σ b and cσd for all a,b,c,d ∈ S. Denote the σ-class containing a ∈ S by aσ. The operation (aσ)(bσ)=(ab)σ is well defined and the quotient S/σ = {aσ | a ∈ S} 2.1. PRELIMINARIES 11

is a semigroup with respect to this operation.

A mapping φ : S → T between two semigroups S and T is a semigroup homomorphism if it preserves the semigroup multiplication. The image of φ is the subsemigroup

im φ = {aφ | a ∈ S} of T , and the kernel of φ is the congruence

ker φ = {(a, b) ∈ S × S | aφ = bφ}

on S. If S = T then φ is an endomorphism of S. If S is any semigroup and σ is any congruence on S then the natural mapping

ν : S → S/σ, a 7→ aσ

is a semigroup homomorphism with kernel σ.

The homomorphism φ is called an embedding if it is injective. In this case the kernel of φ is the trivial congruence

1S = {(a, a) | a ∈ S}, and we say that S embeds in T , and write S . T . If φ is surjective it is called an epimorphism, in which case im φ = T . If φ is bijective then it is called an isomorphism. In this case we say that S is isomorphic to T and write S ∼= T . We think of S as being the same as T . The fundamental homomorphism theorem states that if φ : S → T is a homomorphism then S/ ker φ ∼= im φ. An isomorphism which is also an endomorphism is called an automorphism. Let S and T be any semigroups and σ any congruence on S. Then S is a coextension of T by σ if there is a epimorphism φ : S → T with kernel σ. In this case, by the fundamental homomorphism theorem we have S/σ ∼= T . In the case that S is a group the congruence σ corresponds to a normal subgroup N (the congruence class of the identity) and then S is an extension of N by T in the usual sense that the terminology is used in group theory. The coextension terminology is useful in semigroup theory because congruences rarely correspond to subsemigroups in any meaningful way.

Let S be any semigroup and X any subset of S. The subsemigroup generated by X is

+ hXi = {x1 ··· xn | n ∈ Z , x1,...,xn ∈ X}. Elements of X are called generators of hXi. If every element of X is regular in hXi (resp. idempotent) then hXi is regular-generated (resp. idempotent-generated). If X =

{x1,...,xn} then hXi may be denoted by hx1,...,xni. 2.2. GREEN’S RELATIONS 12

Let X be any set called an alphabet. A word over X is a formal string x1 ··· xn where

x1,...,xn ∈ X. The free semigroup on X is the set of all words

+ FX = {x1 ··· xn | n ∈ Z , x1,...,xn ∈ X} with concatenation

(x1 ··· xn)(y1 ··· ym)= x1 ··· xny1 ··· ym

as the semigroup operation. Elements of X then become generators of FX .

Let R be a subset of FX × FX . The semigroup with (semigroup) presentation hX | Ri is the semigroup

hX | R i = FX /σR

where σR is the smallest congruence on FX which contains R. Elements of R are relations, which are usually thought of as equations which hold in hX | R i and are denoted by a = b rather than (a, b). Elements of hX | R i are congruence classes of words over X but are usually denoted simply by words over X. Two words represent the same element of hX | R i if one can be obtained from the other by applying the relations in R. Consider the following example. Let S = he, f | e2 = e, f 2 = fi be the free semigroup on two idempotent generators. Clearly elements of S can be uniquely represented by words of alternating e’s and f’s. The group with (group) presentation hX | Ri is the group with semigroup presentation hX′ | R′i where

X′ = X ∪{x−1 | x ∈ X}∪{1}

and R′ = R ∪{x1=1x = x, xx−1 = x−1x =1 | x ∈ X}. It will usually be clear from the context whether a presentation is a semigroup presentation or a group presentation.

For a more detailed exposition on semigroup presentations including a proof that every semigroup has a presentation see [32] for example.

2.2. Green’s Relations

Green’s relations are a set of five equivalence relations on a semigroup which were intro- duced by Green [16]. They are useful for describing the structure of a semigroup and when restricted to idempotents motivate the preorders on a boset (or biordered set) introduced by Nambooripad [26, 28]. In this section some useful results are described. Proofs, as well 2.2. GREEN’S RELATIONS 13

as a more detailed exposition on Green’s relations, can be found in [21] Section 1.2 or [22] Chapter 2, for example.

Let S be any semigroup and define Green’s (equivalence) relations, R, L , J , H and D on S by

a R b ⇐⇒ aS1 = bS1, a L b ⇐⇒ S1a = S1b, a J b ⇐⇒ S1aS1 = S1bS1,

H = R ∩ L and D = hR ∪ L i, the smallest equivalence relation on S which contains the union R ∪ L . Observe that R is a left congruence and L is a right congruence. Note that this does not imply that H is a congruence. Denote the R-class (resp. L -class,

J -class, H -class, D-class) containing a ∈ S by Ra (resp. La, Ja, Ha, Da). There are natural partial orders on S/R, S/L , S/J and S/H which are given by

1 Ra ≤ Rb ⇐⇒ a ∈ bS , 1 La ≤ Lb ⇐⇒ a ∈ S b, 1 1 Ja ≤ Jb ⇐⇒ a ∈ S bS ,

Ha ≤ Hb ⇐⇒ La ≤ Lb and Ra ≤ Rb.

The following is an obvious reformulation of the definitions:

a R b ⇐⇒ (∃s, t ∈ S1) b = as, a = bt, a L b ⇐⇒ (∃s, t ∈ S1) b = sa, a = tb, a J b ⇐⇒ (∃s,t,u,v ∈ S1) b = sat, a = ubv, a H b ⇐⇒ (∃s,t,u,v ∈ S1) b = as = ta, a = bu = vb

for all a, b ∈ S.

1 Let e ∈ E and a ∈ Re. Then there exists s ∈ S such that a = es, so ea = ees = es = a.

This shows that e is a left identity for Re and, dually, a right identity for Le. Both of these facts together imply that e is a (right and left) identity for He. This in turn implies the well known fact that there is at most one idempotent in each H -class, for if e2 = e H f = f 2 then e = ef = f. These facts motivate the definitions of boset arrows (see Chapter 3).

A better description of the Green’s relation D is given in the following proposition. A proof can be found in [22] Proposition 2.1.3.

Proposition 2.1. Let S be any semigroup. Then D = R ◦ L = L ◦ R.  2.2. GREEN’S RELATIONS 14

This result shows that for any pair of elements a, b ∈ S which are in the same D-class, there is a pair of elements c,d ∈ S such that a R c L b and a L d R b. This information can be represented in what is commonly referred to as an egg-box,

a ··· c . . . . d ··· b

which is a rectangular array where each row is an R-class, each column is an L -class, and the intersection of each row and column is an H -class.

An important elementary result involving Green’s relations is Green’s Lemma. This result motivates several of the boset axioms (see Chapter 3). A proof can be found in [22] Lemma 2.2.1 and Lemma 2.2.2.

Lemma 2.2. (Green’s Lemma) Let S be any semigroup and a, b ∈ S.

(i) If a R b and s, t ∈ S1 such that b = as and a = bt then the right translations

x 7→ xs and y 7→ yt are mutually inverse R-class preserving bijections from La

onto Lb and Lb onto La respectively; and dually (ii) If a L b and s, t ∈ S1 such that b = sa and a = tb then the left translations x 7→ sx

and y 7→ ty are mutually inverse L -class preserving bijections from Ra onto Rb

and Rb onto Ra respectively. 

The next corollary follows quickly. The proof can be found in [22] Lemma 2.2.3 and involves showing that compositions of appropriate restrictions to H -classes of the right and left translations in Green’s Lemma are bijections between H -classes.

Corollary 2.3. Let S be any semigroup and a, b ∈ S such that a D b. Then |Ha| = |Hb|.

The following is a useful result about the location of products in D-classes. Thought about in the right way, this elementary result forms the simplest case of Pastijn’s general theory of sandwich sets [29], see also Hall [20] Section 3. The proof uses Green’s Lemma and can be found in [21] Theorem 1.2.5.

Proposition 2.4. Let S be any semigroup and a, b ∈ S. Then ab ∈ Ra ∩ Lb if and only if

there is an idempotent e ∈ Rb ∩ La.  2.2. GREEN’S RELATIONS 15

Let a, b ∈ S such that a is regular with inverse a′ ∈ S. If a R b then there exists elements s, t ∈ S1 such that b = as and a = bt. It follows quickly that b is regular with inverse ′ ta . This shows that every element of Ra and, dually, every element of La is regular. Since each R-class in Da intersects nontrivially with each L -class in Da, every element of Da is regular. A regular R-class (resp. L -class, H -class, D-class) is one which contains a regular element, in which case every element in the R-class (resp. L -class, H -class, D-class) is regular.

The following is a useful well known result about inverses in D-classes. The proof can be found in [21] Theorem 1.2.8 and is included here as a good illustration of Green’s relations, the use of Green’s Lemma and the importance of the location of idempotents in determining structural information about a semigroup, the main theme of this thesis.

Proposition 2.5. Let S be any semigroup and a, b ∈ S. Then the H -class Hb contains ′ an inverse a of a if and only if there are idempotents e ∈ Ra ∩ Lb and f ∈ Rb ∩ La, in ′ ′ ′ which case e = aa , f = a a and a is the only inverse of a in Hb.

′ Proof. Suppose there is an inverse a for a in the H -class Hb. The we have the egg-box

a ··· aa′ . . . . a′a ··· b, a′

′ ′ ′ so there are idempotents aa ∈ Ra ∩ Lb and a a ∈ Rb ∩ La. In this case a is the only inverse ′′ of a in Hb, for if a is another then by uniqueness of idempotents in H -classes we have the egg-box diagram

a ··· aa′ = aa′′ . . . . a′a = a′′a ··· b, a′, a′′

and so a′ = a′aa′ = a′′aa′ = a′′aa′′ = a′′.

Conversely, suppose there are idempotents e ∈ Ra ∩ Lb and f ∈ Rb ∩ La. Then by part (ii) ′ of Green’s Lemma since a = af L f there is an element a ∈ Rf ∩ Le = Hb such that the egg-box diagram 2.3. TRANSFORMATION SEMIGROUPS 16

a ··· e = aa′ . . . . f ··· b, a′

holds and we have aa′a = ea = a and a′aa′ = a′e = a′, so a′ is an inverse of a. By uniqueness of idempotents in H -classes we have f = aa′ and we are done. 

The following are well-known results about H -classes which contain idempotents. The first is immediate by Proposition 2.4 and Proposition 2.5 and the second by checking that the bijections in Corollary 2.3 are group homomorphisms.

Proposition 2.6. Let S be any semigroup and e ∈ S an idempotent. Then He is a group.

Proposition 2.7. Let S be any semigroup and e, f ∈ S idempotents such that e D f. Then

He and Hf are isomorphic groups. 

2.3. Transformation Semigroups

Let X be any set. A full transformation of X is a mapping X → X. The full transfor- mation semigroup TX of X is the semigroup of full transformations of X with respect to composition. The full transformation semigroup is important in semigroup theory because of Cayley’s Theorem.

Theorem 2.8. (Cayley’s Theorem) Let S be any semigroup. Then S embeds in TX for some set X. 

The proof is straightforward (see [22] Theorem 1.1.2 for example) and involves checking that the function

ρ : S → TS1 , a 7→ ρa

1 1 where ρa is the transformation S → S , x 7→ xa, is an embedding. The identity element 1 in S is required to ensure that ρ is always injective. The symmetric group SX of X is the subsemigroup of TX which consists of permutations of X. If G is any group then ρg is a permutation for each g ∈ G, so by Cayley’s Theorem G embeds in the symmetric group

SG. The idea of representing an algebra by functions arising from right translation is of fundamental importance in representation theory and becomes particularly subtle when adapted to bosets (see Chapter 4). 2.3. TRANSFORMATION SEMIGROUPS 17

The natural framework for representing bosets uses partial transformations. A partial transformation of X is a mapping A → X where A is some subset of X. If α is a partial transformation of X then denote the domain and image (or range) of α by dom α and im α respectively. The kernel of α is the equivalence relation

ker α = {(x, y) ∈ X × X | x, y ∈ dom α, xα = yα}

on the domain of α and the rank of α is the cardinality

rank α = |im α|

of the image of α. The partial transformation semigroup PTX of X is the semigroup of partial transformations of X with respect to composition wherever possible defined as

follows. If α, β ∈ PTX then the product of α by β is the function

αβ : (im α ∩ dom β)α−1 → X, x 7→ (xα)β

and the image of αβ is (im α ∩ dom β)β.

Idempotents in PTX are partial mappings which are identity mappings on their image.

Green’s relation’s R, L , J , H and D on PTX can be summarised by the following:

D = J

and

α R β ⇐⇒ ker α = ker β, α L β ⇐⇒ im α = im β, α H β ⇐⇒ ker α = ker β and im α = im β, α D β ⇐⇒ rank α = rank β

for any α, β ∈ PTX . This follows by [22] Exercise 2.6.17.

Let X = {1,...,n} and denote any subsemigroup KX of PTX by Kn. An element α ∈ PTn

can be represented by an n-tuple ( x1 ··· xn ) where for each i =1,...,n,

iα if i ∈ dom α xi = ( − otherwise.

Let X be any set. An element σ ∈ SX is a cycle if there is a positive integer n and a

subset Y = {y1,...,yn} of X such that yiσ = yi+1 for i = 1,...,n where yn+1 = y1 and

xσ = x for all x ∈ X \ Y , in which case σ can be represented by the n-tuple ( y1 ··· yn ). The use of n-tuples in these two different ways should be read in context. If σ is any finite 2.3. TRANSFORMATION SEMIGROUPS 18

order permutation then it can be written as a product of disjoint cycles. For example the permutation σ :1 7→ 2, 2 7→ 4, 3 7→ 3, 4 7→ 1, 5 7→ 8, 6 7→ 7, 7 7→ 6, 8 7→ 5

in S8 can be represented by ( 1 2 4 )( 5 8 )( 6 7 ). Note that the cycle ( 3 ) of length 1 is the identity mapping so it can be omitted. This notation is referred to as cycle notation.

The egg-box diagram of PT2 appears in Figure 2.1. In this thesis we use the symbol • to denote idempotents in egg-box diagrams. We draw

and

to mean every idempotent is ≤ the first • and ≥ the second •. All H -classes are trivial, except for the group of units containing the permutations which is cyclic of order 2. No- tice in this example that the group of units is isomorphic to the automorphism group of Figure 2.2 which describes the middle layer of idempotents. The nontrivial automorphism (preserving the arrow types) is achieved by a 180◦ rotation of the diagram. The semigroup

PT2 is an example of a construction called TE introduced in Chapter 4 where E is a boset (see Chapter 3).

The partial transformation semigroup can be disguised as a subsemigroup of a full trans-

formation semigroup in the following way. Let α ∈ PTX and ∞ an element which is not in X. Define a full transformation by

α ∈ TX∪{∞} xα if x ∈ dom α ex 7→ ( ∞ otherwise ∞ 7→ ∞. The function

: PTX → TX∪{∞}, α 7→ α

is an injective semigroup homomorphism. The image of is the subsemigroup of TX∪{∞} which consists of functions whiche fix ∞. The element ∞e essentially means “undefined”, and its purpose is to give the convenience of using full transe formations instead of partial transformations. 2.3. TRANSFORMATION SEMIGROUPS 19

(21) (12) ∼ = S2

(1 − ) (2 − )

(11) (22)

∼ ( − 1) ( − 2) = S1

∼ = S1 ( − − )

Figure 2.1. Egg-box diagram of PT2

Figure 2.2. The middle layer of idempotents in PT2 2.4. FUNDAMENTAL SEMIGROUPS 20

The symmetric inverse semigroup IX of X is the subsemigroup of PTX which consists of partial one-one mappings of X. Idempotents in IX are identity functions

1A : A → X, a 7→ a of subsets A of X. The symmetric inverse semigroup is important in semigroup theory because of the Wagner-Preston Theorem, which is the analogue of Cayley’s Theorem for inverse semigroups. See Wagner [34, 35] and Preston [31].

Theorem 2.9. (The Wagner-Preston Theorem) Let S be any inverse semigroup. Then S

embeds in IX for some set X. 

The proof can be found in [30] Theorem IV.1.6 and involves showing that, for each a ∈ S, the partial mapping −1 ρa : Sa → S, x 7→ xa is injective and the mapping

ρ : S → IS, a 7→ ρa

is an injective homomorphism. Observe that for x ∈ S, xρa is defined and equals the right translate xa if and only if x R xa. This idea, of right translation when Green’s relation R is preserved, is an essential ingredient in this thesis later when we represent bosets using partial transformations.

The egg-box diagram of I3 appears in Figure 2.3. Idempotents appear along the diagonal of each D-class. The H -class containing a rank n idempotent is just the symmetric group

Sn consisting of permutations of {1,...,n}. The group of units is S3 which is isomorphic to the automorphism group of the semilattice of idempotents which appears in Figure 2.4.

The semigroup I3 is an example of Munn’s construction called TE introduced in Section 2.5 where E is a semilattice, which is a special case of a general construction also called

TE introduced in Chapter 4 where E is a boset, motivated by the ideas of Nambooripad (see Chapter 3).

The relationships of set inclusion between the semigroups PTX , TX , IX and SX can be

seen in Figure 2.5. If X is finite then SX is equal to the intersection TX ∩ IX .

2.4. Fundamental Semigroups

In this section the concept of a fundamental semigroup is developed. For details on the theory of fundamental inverse semigroups, developed by Munn [25], see Section 2.5. A fundamental semigroup cannot be shrunk homomorphically without collapsing idempotents 2.4. FUNDAMENTAL SEMIGROUPS 21

(231) (312)

(321) (213) = S3 (123) (132)

(13 − ) (12 − ) (23 − ) (31 − ) (21 − ) (32 − )

(1 − 2) (1 − 3) (2 − 3) (2 − 1) (3 − 1) (3 − 2)

( − 12) ( − 13) ( − 23) ∼= S ( − 21) ( − 31) ( − 32) 2

(2 − − ) (1 − − ) (3 − − )

( − 2 − ) ( − 3 − ) ( − 1 − )

∼ ( − − 1) ( − − 2) ( − − 3) = S1

∼ = S1 ( − − − )

Figure 2.3. Egg-box diagram of I3 2.4. FUNDAMENTAL SEMIGROUPS 22

Figure 2.4. The semilattice E(I3)

PTX

TX IX

SX

Figure 2.5. The relationship between PTX , TX , IX and SX

together. Fundamental semigroups and bosets (see Chapter 3) are good candidates for basic building blocks in semigroup theory because of Corollary 4.8 which says that every semigroup is a biorder preserving coextension of a fundamental semigroup (terminology explained in Chapter 4). We begin with some useful definitions involving idempotents and semigroup congruences.

Let σ be a congruence on a semigroup S. If e is idempotent in S then eσ is idempotent in S/σ. The congruence σ is called idempotent separating if eσ and fσ are different idempotents in S/σ whenever e and f are different idempotents in S, that is,

(∀e, f ∈ E) e σ f ⇒ e = f. 2.4. FUNDAMENTAL SEMIGROUPS 23

It is important to know the answer to the question “does each idempotent aσ in S/σ contain an idempotent in S?” When the answer is yes, there is a chance that structural information embodied in the idempotents of the image will be reflected in the preimage. Clearly the answer is no in general because the trivial semigroup consists of one idempotent and is the image of any semigroup (such as a free semigroup) containing no idempotents.

However, if S is a regular semigroup then the following classical result of Lallement [24] shows the answer is yes, plus a little more.

Lemma 2.10. (Lallement’s Lemma) Let σ be a congruence on a regular semigroup S and aσ an idempotent in S/σ. Then there is an idempotent e ∈ S such that e σ a. Moreover, e can be chosen so that He ≤ Ha. 

The proof can be found in [22] Lemma 2.4.3. It is notable that if (a2)′ is an inverse for a2 then the FitzGerald idempotent e = a(a2)′a has the required property.

A congruence σ on a semigroup S is called an H -congruence if He ≤ Ha whenever e σ a for e ∈ E and a ∈ S. Observe that every H -congruence σ is idempotent separating, for if e, f ∈ E such that e σ f then He ≤ Hf and Hf ≤ He, so e H f, whence e = f.

A semigroup S is fundamental if it cannot be shrunk homomorphically without collapsing idempotents together, that is, if ϕ is a homomorphism from S then

ϕ | E one-one ⇒ ϕ one-one.

This is clearly equivalent to S having no nontrivial idempotent separating congruences.

A nontrivial group G is not fundamental since the universal congruence G × G is idempo- tent separating. At first sight it would appear that the concept of being fundamental is uninteresting from the point of view of group theory. In fact automorphism groups play an essential role in the theory. To illustrate this we can construct a fundamental semigroup from G which is a disguised instance of a subsemigroup of the general construction of TE in Chapter 4. Let M = {1, 0} be the two element monoid and let End(G) be the semigroup of group endomorphisms of G. Define a homomorphism ψ : M → End(G) by

0ψ : G → G, g 7→ 1, 1ψ : G → G, g 7→ g, and form the reverse semidirect product

M ⋉ψ G = {(m, g) | m ∈ M,g ∈ G} 2.4. FUNDAMENTAL SEMIGROUPS 24

1 = G

··· = G

Figure 2.6. Egg-box diagram of G ⊔ G

with semigroup operation (m, g)(n, h)=(mn, gnh) where gn = g(nψ).

This semigroup can be identified with the disjoint union of G with a right zero semigroup G = {g | g ∈ G}. Make the identification

G ⊔ G ≡ M ⋉ψ G where g ≡ (1,g) and g ≡ (0,g) for each g ∈ G. The semigroup operation · on G ⊔ G, induced by the semidirect product multiplication, is given by, for g, h ∈ G,

g · h = gh, g · h = gh, g · h = g · h = h,

where group multiplication in G is denoted by juxtaposition.

The egg-box diagram of G ⊔ G appears in Figure 2.6. The subsemigroup G is a group in which only one element is idempotent, and the subsemigroup G is a right zero semigroup in which every element is idempotent. The following is a direct proof that G ⊔ G is fundamental which is included to motivate the proof technique which is gradually adapted to a variety of settings. The result is a consequence of Theorem 5.3, a very general umbrella result below. 2.4. FUNDAMENTAL SEMIGROUPS 25

Proposition 2.11. Let G be any group. Then G ⊔ G is fundamental.

Proof. Let σ be an idempotent separating congruence on G ⊔ G and g, h ∈ G. We show that σ is trivial by considering several cases. If g σ h then g = h since σ is idempotent separating. If g σ h then g = 1 · g σ 1 · h = h, which implies g = h, so g = h. Finally, if g σ h then

1= g · g−1 σ h · g−1 = hg−1

which is a contradiction since σ is idempotent separating. This shows that σ is trivial. Hence G ⊔ G is fundamental. 

We finish this section with two classical examples of fundamental semigroups. Both semi- groups are fundamental by Theorem 5.3, but direct proofs will be provided by modifying the previous proof. Consider the full transformation semigroup TX on a set X. A partial egg-box diagram of TX appears in Figure 2.7. The middle section of the egg-box diagram has been removed. Note the similarity between this diagram and the egg-box diagram for G ⊔ G in Figure 2.6. The egg-boxes form a chain under the ordering of D-classes.

The highest D-class is the symmetric group SX and the lowest D-class consists of rank 1 mappings, which are constant mappings

κx : X → X, y 7→ x,

which play an important role in the proof that TX is fundamental.

Proposition 2.12. Let X be any set. Then TX is fundamental.

Proof. Let σ be an idempotent separating congruence on TX and α, β ∈ TX such that α σ β. Then, for each x ∈ X,

κxα = κxα σ κxβ = κxβ.

This implies that κxα = κxβ since σ is an idempotent separating congruence, which in turn implies that xα = xβ since κxα and κxβ are equal constant mappings. This proves that

α = β, which implies that σ is the trivial congruence. Hence TX is fundamental. 

Now consider the symmetric inverse semigroup IX on a set X. A partial egg-box diagram of IX appears in Figure 2.8. The highest D-class is the symmetric group SX and the lowest two consist of the empty mapping 1∅ and the rank 1 mappings 1x,y : {x}→{y}, x 7→ y.

Write 1x for 1x,x for each x ∈ X. These functions play an important role in the proof that

IX is fundamental. 2.5. THE MUNN SEMIGROUP 26

1X = SX

. .

constant ··· ← mappings

Figure 2.7. Partial egg-box diagram of TX

Proposition 2.13. Let X be any set. Then IX is fundamental.

Proof. Let σ be an idempotent separating congruence on IX and α, β ∈ IX . Suppose α σ β. It is sufficient to prove xα = xβ for each x ∈ dom α, for then, by symmetry α = β, which shows σ is trivial, whence IX is fundamental. Let x ∈ dom α. Then

1x if xα = xβ 1x =1xα1xα,x σ 1xβ1xα,x = ( 1∅ otherwise.

Since σ is idempotent separating we must have xα = xβ and we are done. 

2.5. The Munn Semigroup

In this section evidence is provided that fundamental semigroups are a good choice for a basic building block in semigroup theory. The results are due to Munn [25], who proves that every inverse semigroup is a coextension of a fundamental inverse semigroup with the same semilattice of idempotents, and also that every fundamental inverse semigroup can be reconstructed from its semilattice of idempotents. These results are special cases of more general results, involving bosets (see Chapter 3), to be proved later in this thesis. Proofs for the inverse semigroup case can be found in [22], Section 5.4, for example. 2.5. THE MUNN SEMIGROUP 27

1X = SX

. .

rank 1 ← mappings . ..

empty ← mapping

Figure 2.8. Partial egg-box diagram of IX 2.5. THE MUNN SEMIGROUP 28

Let E be any poset and e ∈ E. The principal ideal of E generated by e is the set ω(e)= {x ∈ E | x ≤ e} and a principal ideal isomorphism of E is a bijection α : ω(e) → ω(f) where e, f ∈ E and both α and α−1 preserve the partial order on E, that is, (∀e, f ∈ E) e ≤ f ⇐⇒ eα ≤ fα.

Let E be any semilattice. The Munn semigroup TE is the subsemigroup of IE defined by

TE = hα | α is a principal ideal isomorphism of Ei. Let α : ω(e) → ω(f) and β : ω(g) → ω(h) be principal ideal isomorphisms of E. Then the product of α by β is the principal ideal isomorphism αβ : ω((fg)α−1) → ω((fg)β), x 7→ (xα)β, since dom(αβ) = (im α ∩ dom β)α−1 =(ω(f) ∩ ω(g))α−1 =(ω(fg))α−1 = ω((fg)α−1) and im (αβ) = (im α ∩ dom β)β =(ω(f) ∩ ω(g))β =(ω(fg))β = ω((fg)β). This allows us to replace the angular brackets with curly brackets, so

TE = {α | α is a principal ideal isomorphism on E}. It is usual to define the Munn semigroup using curly brackets, and verify that the set is closed under multiplication. Our definition above using angular brackets motivates the

definition of TE when E is any boset (see Chapter 4).

Idempotents in TE are identity principal ideal isomorphisms,

1ω(e) : ω(e) → ω(e), x 7→ x.

The importance of TE will be illustrated with the following example. Let X be any set and

consider the symmetric inverse semigroup IX of X. Denote the semilattice of idempotents

of IX by E.

Idempotents in IX are identity mappings 1A : A → A, a 7→ a on subsets A of X and

1A ≤ 1B ⇐⇒ A ⊆ B.

For each A ⊆ X the principal ideal of E generated by 1A is the set

ω(1A)= {1B | B ⊆ A} 2.5. THE MUNN SEMIGROUP 29

and ∼ ω(1A) = ω(1B) ⇐⇒ |A| = |B|.

Idempotents in TE are identity principal ideal isomorphisms 1ω(1A).

Define a mapping

φ : IX → TE, α 7→ α where α is the principal ideal isomorphism

α : ω(1dom α) → ω(1im α), 1A 7→ 1Aα. It is easy to check that αβ = αβ so φ is a semigroup homomorphism. It is also easy to ∼ check that φ is one-one and onto. Hence IX = TE, so the fundamental semigroup IX can be reconstructed from its semilattice of idempotents. It is immediate that IX and TE have isomorphic semilattices of idempotents, so TE is an inverse semigroup with semilattice of idempotents isomorphic to E. These observations are examples of general results due to Munn [25]. The proofs of these results appear in [22], Section 5.4 for example.

Theorem 2.14. Let E be any semilattice. Then the Munn semigroup TE is an inverse semigroup with semilattice of idempotents isomorphic to E. 

As in the above example, the proof in [22] Theorem 5.4.1 involves showing that the function

E → E(TE), e 7→ 1ω(e) is a semilattice isomorphism. The Munn semigroup TE is inverse since it is regular with a semilattice of commuting idempotents.

Munn semigroups are important because of the following results, which show how they can be used to construct all fundamental inverse semigroups from semilattices.

Theorem 2.15. Let S be any inverse semigroup with semilattice of idempotents E. Then there is a semigroup homomorphism φ : S → TE with kernel µ = {(a, b) ∈ S × S | (∀e ∈ E) a−1ea = b−1eb} which is the maximum idempotent separating congruence on S. 

The proof can be found in [22], Theorem 5.4.4. The homomorphism φ is given by a 7→ φa where φa is the principal ideal isomorphism

−1 −1 −1 φa : ω(aa ) → ω(a a), x 7→ a xa. The maximality of µ clearly implies that S/µ is fundamental. Thus the image of φ is the fundamental inverse semigroup im φ =∼ S/µ. This result shows that every inverse 2.5. THE MUNN SEMIGROUP 30

semigroup is a coextension of a fundamental inverse semigroup with the same semilattice of idempotents. This last statement was generalised to all semigroups by Easdown [9], discussed below as Corollary 4.8. If S is fundamental then φ is one-one. It is not necessarily onto, but the image of φ is a full subsemigroup of TE.

Theorem 2.16. Let S be any inverse semigroup with semilattice of idempotents E. Then

S is fundamental if and only if it is isomorphic to a full subsemigroup of TE. 

The proof can be found in [22], Theorem 5.4.5. This result shows that every fundamental inverse semigroup can be reconstructed from its semilattice of idempotents. The Munn semigroup TE is the maximum fundamental inverse semigroup with semilattice of idempo- tents isomorphic to E.

These results have been generalised by various authors, using different constructions, to the class of regular semigroups. Hall [20] uses idempotent generated semigroups and a generalisation of the homomorphism φ to regular semigroups. Grillet [18, 19] uses posets with cross-connections. Nambooripad [26, 28] uses regular bosets, which are the systems of idempotents of regular semigroups, discussed in Chapter 3. Clifford [1] uses the concept of a warp, which is a special partial groupoid.

It is the aim of this thesis to generalise these results further to larger classes of semigroups. The approach is to use a further generalisation of the homomorphism φ to arbitrary semi- groups, as well as a general construction called TE where E is an arbitrary boset, appearing in Jordan’s thesis [23].

A regular semigroup always possesses a maximum idempotent separating congruence ([22] Theorem 2.4.5). The situation in general is far from clear. By Zorn’s Lemma a semigroup always possesses a maximal idempotent separating congruence, but this need not be max- imum, nor bear any obvious relationship to the arrows and between idempotents introduced above in Section 2.1. To illustrate this consider again the free semigroup

S = he, f | e2 = e, f 2 = fi

on two idempotent generators. Define two congruences σ and τ on S by

a σ b ⇐⇒ a and b end with the same letter

and a τ b ⇐⇒ a and b begin with the same letter. Both quotients S/σ and S/τ are two element bands which can be described by the diagrams 2.5. THE MUNN SEMIGROUP 31

eσ fσ and

eτ fτ respectively. Both σ and τ are maximal idempotent separating congruences, since the only congruence which is strictly larger than either of them is the universal congruence S × S. This shows that there is no maximum idempotent separating congruence on S, for if µ is the maximum idempotent separating congruence then it must contain both σ and τ, so eµfeµf, since eσfe and fτfe, which contradicts the fact that µ is idempotent separating. CHAPTER 3

Bosets

In this chapter we introduce Nambooripad’s concept of a boset, which abstractly describes the system of idempotents of an arbitrary semigroup. Another good exposition on the topic can be found in [21] Section 1.5 and Chapter 3. The terminology boset was coined by Jordan [23] to serve both as an abbreviation of biordered set, due to Nambooripad [26, 28], and as a generalisation of poset, itself an abbreviation of partially ordered set.

3.1. A Vector Space Example

We begin by motivating the topic with an example involving linear operators of vector spaces which specialises to matrices in the finite dimensional case.

Subspaces of a vector space form a poset under set inclusion ⊆ and also under reverse set inclusion ⊇. We can combine both of these in the following apparently novel way, which we will see in fact captures precisely the relationship between idempotent linear operators.

Let V be a vector space and

E = E(V )= {(A, B) | V = A ⊕ B} where V = A ⊕ B means A, B ≤ V , A + B = V and A ∩ B = {0} (internal direct sum). Define a partial operation ⋆ on E by

(A, B) ⋆ (C,D)=(A +(B ∩ C), (B + C) ∩ D).

There is no hope of the domain of ⋆ being E×E in general because of the following:

Let V = Z2 ⊕ Z2 = hv,wi. The lattice of subspaces of V is given by Figure 3.1. Put A = D = hvi, B = hwi and C = hv + wi. Then (A, B), (C,D) ∈E but

(A +(B ∩ C), (B + C) ∩ D)=(A, A) ∈E/ .

However, E becomes a partial groupoid with respect to ⋆ if we define the domain to be

DE = {((A, B), (C,D)) ∈E×E| A ⊇ C, A ⊆ C, B ⊆ D or B ⊇ D}. 32 3.1. A VECTOR SPACE EXAMPLE 33 V

hvi hwi hv + wi

{0}

Figure 3.1. The lattice of subspaces of V = Z2 ⊕ Z2

Define arrow relations and on E by

(A, B) (C,D) ⇐⇒ A ⊇ C, (A, B) (C,D) ⇐⇒ B ⊆ D. Then

DE = {(α, β) ∈E×E| α β, α β, α β or α β}. The general formula for ⋆ then can easily be seen to simplify as follows:

(i) if α =(A, B) β =(C,D) then α ⋆ β =(A, (B + C) ∩ D) and β ⋆ α = α,

(ii) if α =(A, B) β =(C,D) then α ⋆ β =(A +(B ∩ C),D) and β ⋆ α = β.

For example, it is straightforward to check from the definitions of and , and using the modular law C ≤ A ⇒ A ∩ (B + C)=(A ∩ B)+ C for the subspace lattice of V , that V = A ⊕ ((B + C) ∩ D) in case (i), and V =(A +(B ∩ C)) ⊕ D in case (ii), so that ⋆ is sensibly defined.

The relations and inherit reflexivity and transitivity from ⊇ and ⊆. Neither are antisymmetric because a fixed subspace may have many different complements. Together, 3.1. A VECTOR SPACE EXAMPLE 34 however, they possess an antisymmetric property in the following sense. Suppose (A, B) (C,D) (A, B). Then A ⊇ C, B ⊇ D and V = A ⊕ B = C ⊕ D. Let a ∈ A. Then a = c + d for some c ∈ C and d ∈ D. This implies that d = a − c ∈ A ∩ B = {0}, which in turn implies a = c ∈ C, so A ⊆ C. Together with A ⊇ C, this implies that A = C. By symmetry B = D, so (A, B)=(C,D).

We have proven the intertwined antisymmetric property: (A, B) (C,D) (A, B) ⇒ (A, B)=(C,D). In particular, the relation = ∩ becomes a partial order.

Notice that β β ⇒ α α α ⋆ β and α α ⇒ β β α ⋆ β.

These two implications hold in a general boset setting (below), and when combined with transitivity of arrows they yield two of the most important ways of building arrow diagrams.

Now consider the semigroup S = End(V )= {linear operators : V → V } under composition of linear transformations, read from left to right. Recall that E = E(S)= {e ∈ S | e2 = e}. Observe that if e ∈ E then V = ker e ⊕ im e so that the following function is sensibly defined: θ : E →E, e 7→ (ker e, im e). It is straightforward to verify that θ is one-one and onto, and, for all e, f ∈ E, e f ⇐⇒ eθ fθ, 3.1. A VECTOR SPACE EXAMPLE 35

in which case (ef)θ =(eθ) ⋆ (fθ), and dually e f ⇐⇒ eθ fθ, in which case (fe)θ =(fθ) ⋆ (eθ). This shows θ is a boset isomorphism in the sense explained below.

Let V be a vector space over a field F with basis B. We define a subset F of E for which = = so F is a poset under the inherited arrow operation. In fact, F is a disguised instance of the semilattice P(B) of subsets of B under set inclusion. Define F = F(V )= {(hB \ Xi, hXi) | X ∈ P(B)}. Then F is a subset of E since V = hB \ Xi⊕hXi for any subset X of B. Observe that α =(hB \ Xi, hXi) β =(hB \ Y i, hY i) ⇐⇒ hB\ Xi⊇hB\ Y i ⇐⇒ B\ X ⊇B\ Y ⇐⇒ X ⊆ Y ⇐⇒ hXi⊆hY i ⇐⇒ α β so F is a poset under = = . Also, the operation ⋆ inherited from E defined at the beginning of this section is full on F and, as is easily checked, simplifies to (hB \ Xi, hXi) ⋆ (hB \ Y i, hY i)=(hB \ (X ∩ Y )i, hX ∩ Y i). The mapping F → P(B), (hB \ Xi, hXi) 7→ X is a semilattice isomorphism.

Now consider a B × B matrix

M = [mij]i,j∈B. Call M row-finite if each row contains finitely many nonzero entries, that is

(∀i ∈ B) |{mij | j ∈ B, mij =06 }| < ∞, and column-finite if each column contains finitely many nonzero entries, that is

(∀j ∈ B) |{mij | i ∈ B, mij =06 }| < ∞. 3.1. A VECTOR SPACE EXAMPLE 36 01000 00000   00002    10000     00200      Figure 3.2. Example rook matrix in Rook5(Z3)

Put

MatB, row(F)= {row-finite B × B matrices over F} and

MatB, col(F)= {column-finite B × B matrices over F} where the semigroup operation is matrix multiplication. Then ∼ End(V ) = MatB, row(F)

and, dually, ∗ ∼ End (V ) = MatB, col(F). Define the rook monoid

S = RookB(F)= {B×B matrices over F | each row and column contains at most one nonzero entry}.

The rook monoid is a submonoid of

MatB, row(F) ∩ MatB, col(F).

If V is an n-dimensional vector space we may write Rookn(F) instead of RookB(F). El- ements of S are called rook matrices. The nonzero entries in each rook matrix can be thought of as rooks on a B × B chess board in nonattacking positions. The rook terminol- ogy is due to Solomon [33] who considered rook matrices in which every nonzero entry is equal to 1. For an example of a rook matrix, see Figure 3.2. Idempotents in S are rook matrices in which every nonzero entry is equal to 1 and appears along the main diagonal. The mapping

ψ : S → IB, A = [aij]i,j∈B 7→ ψA : i 7→ j if aij =06 is a semigroup epimorphism and the quotient ∼ S/ ker ψ = IB

is isomorphic to the monoid considered by Solomon [33]. Note that the mapping ψA is the partial one-one mapping associated with the matrix obtained from A by replacing each 3.2. A GROUP EXAMPLE 37

nonzero entry with 1 restricted to the subset of B which lies outside of the kernel. Observe that if e = [eij]i,j∈B ∈ E(S) and θe is the associated idempotent linear operator then

ker θe = hB \ Xi and im θe = hXi where

X = {i ∈B| eii =06 } so that the following function is sensibly defined:

θ : E →F, e 7→ (ker θe, im θe).

It is straightforward to verify that θ is one-one and onto, and, for all e, f ∈ E,

e f ⇐⇒ eθ fθ.

This shows θ is a semilattice isomorphism. Thus ψ may be regarded as the representation of S by its TE whose kernel is the maximum idempotent separating congruence on S.

3.2. A Group Example

Since every vector space is an abelian group under addition, one might ask if this idea can be extended to arbitrary groups. This can be done in the following way:

Let G be a group and E = E(G)= {(A, B) | G = A ⋊ B} where G = A ⋊ B means A E G, B ≤ G, G = AB and A ∩ B = {1} (internal semidirect product). Note that this definition is one-sided because B may not be a normal subgroup of G.

Define a partial operation ⋆ on E by

(A, B) ⋆ (C,D)=(A(B ∩ C), (BC) ∩ D).

If G is abelian then the semidirect product decompositions are direct and one can write this rule additively

(A, B) ⋆ (C,D)=(A +(B ∩ C), (B + C) ∩ D).

A special case is the vector space construction above. The vector space example Z2 ⊕ Z2 shows the domain of ⋆ cannot be E×E in general. However, somewhat surprisingly, always A(B ∩ C) E G by the following lemma. 3.2. A GROUP EXAMPLE 38

Lemma 3.1. Let G be any group such that

G = N ⋊ H = M ⋊ K.

Then N(H ∩ M) E G.

Proof. It is sufficient to check

(H ∩ M)g ⊆ N(H ∩ M)

for all g ∈ G, for then

(N(H ∩ M))g = N g(H ∩ M)g ⊆ N(N(H ∩ M)) ⊆ N(H ∩ M)

for all g ∈ G since N E G, and we will be done. Let x ∈ H ∩ M and g ∈ G, so

(∃h ∈ H)(∃n ∈ N) g = hn.

Then −h xg = xhn = n−1xhnx−hxh = n−1nx xh ∈ N(H ∩ M)

−h since n−1nx ∈ N, as N E G, and xh ∈ H ∩ M, as x, h ∈ H and M E G. 

Again E becomes a partial groupoid with respect to ⋆ if we define the domain to be

DE = {(α, β) ∈E×E| α β, α β, α β or α β} where

(A, B) (C,D) ⇐⇒ A ⊇ C (A, B) (C,D) ⇐⇒ B ⊆ D.

Again, with some work, the general formula for ⋆ then simplifies as follows:

(i) if α =(A, B) β =(C,D) then

α ⋆ β =(A, (BC) ∩ D) and β ⋆ α = α,

(ii) if α =(A, B) β =(C,D) then

α ⋆ β =(A(B ∩ C),D) and β ⋆ α = β.

As before, it is straightforward to check from the definitions of and , and using the modular law C ≤ A ⇒ A ∩ (BC)=(A ∩ B)C for the subgroup lattice of G, that

G = A ⋊ ((BC) ∩ D) 3.2. A GROUP EXAMPLE 39

in case (i), and, in light of Lemma 3.1, G =(A(B ∩ C)) ⋊ D in case (ii), so that ⋆ is sensibly defined.

Again, the relations and inherit transitivity from ⊇ and ⊆. Neither are antisym- metric, but together they possess the intertwined antisymmetric property: (A, B) (C,D) (A, B) ⇒ (A, B)=(C,D). In particular, the relation = ∩ becomes a partial order.

Notice again, that β β ⇒ α α α ⋆ β

and α α ⇒ β β α ⋆ β.

Now consider the semigroup S = End(G)= {group homomorphisms : G → G} under composition of mappings, read from left to right. Recall that E = E(S)= {e ∈ S | e2 = e}. Observe that if e ∈ E then G = ker e ⋊ im e so that the following mapping is sensibly defined: θ : E →E, e 7→ (ker e, im e). It is straightforward to verify that θ is one-one and onto, and, for all e, f ∈ E, e f ⇐⇒ eθ fθ, in which case (ef)θ =(eθ) ⋆ (fθ), and dually, e f ⇐⇒ eθ fθ, 3.2. A GROUP EXAMPLE 40

G = AB

A B

{1} = A ∩ B

Figure 3.3. Sublattice of the lattice of subgroups when G = A ⋊ B

G

hai hbi habi

{1}

Figure 3.4. Lattice of subgroups of G = C2 × C2

in which case (fe)θ =(fθ) ⋆ (eθ).

This shows θ is a boset isomorphism in a sense explained below.

Using this group-theoretic construction we can build arrow diagrams associated with the lattices of subgroups, because (A, B) ∈ E when the Hasse diagram in Figure 3.3 holds, where • denotes a normal subgroup and ◦ denotes a subgroup that need not be normal.

We illustrate this idea using several examples. First, consider the direct product of cyclic

groups G = C2 × C2 = ha, bi. The lattice of subgroups of G is in Figure 3.4 and the arrow diagram of E is in Figure 3.5.

3 2 b 2 Next consider the symmetric group G = S3 = ha, b | a = b = 1, a = a i. The lattice of subgroups of G is in Figure 3.6 and the arrow diagram of E is in Figure 3.7. 3.2. A GROUP EXAMPLE 41

({1},G)

(hai, hbi) (hai, habi)

(habi, hbi) (hbi, habi)

(habi, hai) (hbi, hai)

(G, {1})

Figure 3.5. Arrow diagram of E(C2 × C2)

G

hai hbi habi ha2bi

{1}

Figure 3.6. Lattice of subgroups of G = S3 3.2. A GROUP EXAMPLE 42

({1},G)

(hai, hbi) (hai, ha2bi) (hai, habi)

(G, {1})

Figure 3.7. Arrow diagram of E(S3)

G

2 hai ha , bi ha2, abi

ha2i hbi habi ha2bi ha3bi

{1}

Figure 3.8. Lattice of subgroups of G = D8 3.2. A GROUP EXAMPLE 43 ({1},G)

(ha2, bi, habi) (ha2, bi, ha3bi)

(ha2, abi, hbi) (ha2, abi, ha2bi)

(hai, hbi) (hai, habi) (hai, ha2bi) (hai, ha3bi)

(G, {1})

Figure 3.9. Arrow diagram of E(D8)

4 2 b 3 Finally, consider the dihedral group G = D8 = ha, b | a = b = 1, a = a i which is the group of symmetries of the square. The lattice of subgroups of G is in Figure 3.8 and the arrow diagram of E is in Figure 3.9. We will now recover the arrows of E by computing directly with endomorphisms of D8. Let [x, y] denote the endomorphism of D8 induced by a 7→ x, b 7→ y. Only choices of x, y which satisfy the relations in the presentation

4 2 b 3 D8 = ha, b | a = b =1, a = a i are allowed. It is not hard to check that there are exactly 36 distinct choices giving rise to endomorphisms, of which 10 are idempotents. The arrow relationships between idempotents in E = E(End(D8)) can be seen in Figure 3.10. The boset corresponding to E (and E) is regular in the sense of Nambooripad (see Section 3.8). However, the semigroup

End(D8) is not regular as the following proposition shows. The entire egg-box diagram of

End(D8), which has several nonregular D-classes, appears in Figure 3.11. 3.2. A GROUP EXAMPLE 44 [a, b]

[ab, 1] [a3b, 1]

[b, b] [a2b, a2b]

[1, b] [1, ab] [1, a2b] [1, a3b]

[1, 1]

Figure 3.10. Arrow diagram of E(End(D8))

Proposition 3.2. Let D2n be the dihedral group

ha, b | an = b2 =1, ab = a−1i for any integer n ≥ 3. Then End(D2n) is regular if and only if n is a product of distinct primes.

Proof. First, suppose n is a product of distinct primes. We prove that End(D2n) is regular.

Let ϕ ∈ End(D2n). Then ϕ is determined by one of the following for some i, j ∈{1,...,n}:

(i) ϕ : a 7→ ai, b 7→ ajb, (ii) ϕ : a 7→ aib, b 7→ ajb, (iii) ϕ : a 7→ aib, b 7→ aj, (iv) ϕ : a 7→ ai, b 7→ aj.

It is sufficient to construct an element ψ ∈ End(D2n), in each case, such that

a(ϕψϕ)= aϕ and b(ϕψϕ)= bϕ 3.2. A GROUP EXAMPLE 45

1 = [a, b] α = [a, ab] α2 = [a, a2b] α3 = [a, a3b] β = [a3, b] αβ = [a3, a3b] α2β = [a3, a2b] α3β = [a3, ab]

k ∼ hα, βi = D8

γ γα δ δα δα2 δα3 k k k k k k [a2, b] [a2, ab] [a2b, a2] [a3b, a2] [b, a2] [ab, a2]

γα2 =[a2, a2b] [a2, a3b] = γα3 [a2b, b] [a3b, ab] [b, a2b] [ab, a3b] k k k k k k αγ αγα αδ αδα αδα2 αδα3

ǫ ǫα ǫα2 ǫα3 k k k k

βǫ = α2ǫ = [b, 1] [ab, 1] [a2b, 1] [a3b, 1]

βαǫ = αǫ = [b, b] [ab, ab] [a2b, a2b] [a3b, a3b]

βχ = αχ = [1, b] [1, ab] [1, a2b] [1, a3b]

k k k k χ χα χα2 χα3

[1, a2]

[a2, a2]

[a2, 1]

[1, 1]

Figure 3.11. Egg-box diagram of End(D8) 3.2. A GROUP EXAMPLE 46 for then ϕψϕ = ϕ since ϕ and ψ are group homomorphisms. Observe that

aib if k is odd (aib)k = ( 1 if k is even. This will often be used without comment.

(i) ϕ : a 7→ ai, b 7→ ajb.

Let d be the greatest common divisor of i and n. Then there are integers k and m such that i = dk and n = md. Since n is a product of distinct primes, and any divisor of m is also a divisor of n, the greatest common divisor of i and m must be 1. By the Euclidean Algorithm there are integers x and y such that

xi + ym =1.

Define ψ by ψ : a 7→ ax, b 7→ a−xjb. Then

((aϕ)ψ)ϕ =(aiψ)ϕ = axiϕ = a1−ymϕ = ai−ymdk = ai−ynk = ai = aϕ

and ((bϕ)ψ)ϕ = ((ajb)ψ)ϕ =(axja−xjb)ϕ = bϕ. (ii) ϕ : a 7→ aib, b 7→ ajb. We must have n = 2m for some odd integer m since n is a product of distinct primes and (aib)n =(aϕ)n = anϕ =1ϕ =1. Also,

1=1ϕ =(abϕ)(aϕ)=[(bϕ)(aϕ)]2 =(ajbaib)2 = a2(j−i)

so i − j =0 or m (mod n). It is sufficient to consider the following subcases: (1) ϕ : a 7→ aib, b 7→ aib, (2) ϕ : a 7→ aib, b 7→ ai+mb. 3.2. A GROUP EXAMPLE 47

(1) ϕ : a 7→ aib, b 7→ aib. If i is odd define ψ by ψ : a 7→ ai+m, b 7→ b. Then i(i + m) is even so ((aϕ)ψ)ϕ = ((aib)ψ)ϕ =(ai(i+m)b)ϕ = bϕ = aϕ and ((bϕ)ψ)ϕ = ((aib)ψ)ϕ = bϕ. Otherwise, i is even and (aib)ϕ = bϕ = aib, so ϕ = ϕ2. Put ψ = ϕ. Then ϕψϕ = ϕ3 = ϕ. (2) ϕ : a 7→ aib, b 7→ ai+mb. If i is odd then i + m is even, so aϕ3 =(aib)ϕ2 =(aibai+mb)ϕ = amϕ = aib = aϕ and bϕ2 =(ai+mb)ϕ = bϕ. Put ψ = ϕ. Then ϕψϕ = ϕ3 = ϕ. Otherwise i is even and i + m is odd, so aϕ4 =(aib)ϕ3 =(ai+mb)ϕ2 =(aibai+mb)ϕ = amϕ = aϕ and bϕ4 =(ai+mb)ϕ3 =(aibai+mb)ϕ2 = amϕ2 =(aib)ϕ = bϕ. Put ψ = ϕ2. Then ϕψϕ = ϕ4 = ϕ. (iii) ϕ : a 7→ aib, b 7→ aj. As before, in case (ii), we must have n =2m for some odd integer m. Also, a2j =(bϕ)2 = b2ϕ =1ϕ =1, so j =0 or m (mod n). Again, there are two subcases to consider: (1) ϕ : a 7→ aib, b 7→ 1, (2) ϕ : a 7→ aib, b 7→ am. 3.2. A GROUP EXAMPLE 48

(1) ϕ : a 7→ aib, b 7→ 1. Define ψ by ψ : a 7→ 1, b 7→ ab. Then ((aϕ)ψ)ϕ = ((aib)ψ)ϕ =(ab)ϕ = aϕ and ((bϕ)ψ)ϕ =1= bϕ. (2) ϕ : a 7→ aib, b 7→ am. If i is odd define ψ by ψ : a 7→ b, b 7→ amb. Then ((aϕ)ψ)ϕ = ((aib)ψ)ϕ =(bamb)ϕ = amϕ = aϕ and ((bϕ)ψ)ϕ =(amψ)ϕ = bϕ. Otherwise i is even, so aϕ3 =(aib)ϕ2 = bϕ2 = amϕ = aϕ and bϕ3 = amϕ2 = aϕ2 =(aib)ϕ = bϕ. Put ψ = ϕ. Then ϕψϕ = ϕ3 = ϕ. (iv) ϕ : a 7→ ai, b 7→ aj. We must have 1=(bϕ)2 = [(ab)ϕ]2 = a2j = a2(i+j) = a2i, so either aϕ = bϕ =1 or aϕ, bϕ ∈{1, am} where n =2m for some odd integer m. There are four subcases to consider: (1) ϕ : a 7→ 1, b 7→ 1, (2) ϕ : a 7→ am, b 7→ 1, (3) ϕ : a 7→ am, b 7→ am, (4) ϕ : a 7→ 1, b 7→ am, where n =2m for some odd integer m in cases (2)–(4). For cases (1)–(3) we have ϕ2 = ϕ, so put ψ = ϕ. Then ϕψϕ = ϕ3 = ϕ. 3.3. THE BOSET OF A SEMIGROUP 49

(4) ϕ : a 7→ 1, b 7→ am. Define ψ by ψ : a 7→ b, b 7→ 1. Then ((aϕ)ψ)ϕ =1= aϕ and ((bϕ)ψ)ϕ =(amψ)ϕ = bϕ.

This completes the proof that End(D2n) is regular.

We now prove that n is a product of distinct primes if End(D2n) is regular. We use a contrapositive argument. Suppose n is not a product of distinct primes. Then there is a 2 prime p and an integer m such that n = p m. We show that End(D2n) has a nonregular element, for then End(D2n) is not regular and we will be done. Define ϕ ∈ End(D2n) by ϕ : a 7→ apm, b 7→ b

k and let ψ ∈ End(D2n). If aψ = a for some k ∈{1,...,n} then

2 2 ((aϕ)ψ)ϕ =(apmψ)ϕ = akpmϕ = akp m = akmn =1 =6 apm = aϕ.

Otherwise aψ = akb for some k ∈{1,...,n} and

(akb)ϕ = akmpb =6 aϕ if pm is odd ((aϕ)ψ)ϕ =(apmψ)ϕ = ( 1ϕ =1 =6 aϕ if pm is even.

This shows that ϕψϕ =6 ϕ for all ψ ∈ End(D2n), so ϕ is not regular. 

3.3. The Boset of a Semigroup

In this section we introduce Nambooripad’s concept of the boset of idempotents of an arbitrary semigroup (see Theorem 1.1 (a1) of [28]).

Let S be any semigroup and E its set of idempotents. Observe by the remarks preceding Proposition 2.1 that if e, f ∈ E then

e R f if and only if ef = f and fe = e, and dually, e L f if and only if ef = e and fe = f. 3.3. THE BOSET OF A SEMIGROUP 50

Define arrow relations and on E, which are half of Green’s relations R and L respectively, by e f ⇐⇒ fe = e, e f ⇐⇒ ef = e. We also write e f if f e and e f if f e, and arrows can be combined so that

= ∩ = R | E,

= ∩ = L | E, = ∩ = ≤,

where ≤ is the natural partial order on E, which is given by

e ≤ f ⇐⇒ ef = fe = e.

The arrow relations and are reflexive and transitive, and together they possess an intertwined antisymmetric property:

e f e ⇒ e = f.

If the arrows coincide then E is a poset under = = .

The set E also inherits a partial multiplication from S. The multiplication is not full, in general, since the product of two idempotents may not be idempotent. We are however guaranteed idempotent products in the following cases. If e f then ef = eef = efef is idempotent and we have f

fe = e ef,

and dually, if e f then fe is idempotent and we have f

ef = e fe.

The boset of the semigroup S is the set E, together with the arrow relations and , and the partial multiplication inherited from S over the domain

DE = ∪ ∪ ∪ . Informally, we can think of the boset as the skeleton of the semigroup, which consists of the idempotents, arrows and products due to arrows. 3.3. THE BOSET OF A SEMIGROUP 51

A natural question to ask is “what properties of the semigroup remain when attention is paid only to its boset?” Since the partial multiplication on E is inherited from S and the arrows are half of Green’s relations R and L we may expect some remnants of the properties of associativity and Green’s relations. We now describe some properties of bosets which form the basis for Nambooripad’s boset axioms.

Let e,f,g ∈ E. If

e

f g

then the products fe and ge are defined in the boset and (in the semigroup) (fe)(ge) = f(eg)e = (fg)e = fe, so fe ge. We can think of this as an arrow analogue of the fact that L is a right congruence on a semigroup.

If e f g then (eg)f =(e(gf)) = ef and if

e

f g

then (ge)(fe)=(g(ef))e =(gf)e. These two properties can be thought of as remnants of associativity.

If

e

f g

fe ge

then the product fg = ffg = fefg = fegefg = fgfg is idempotent and we have fg g, fg e and (fg)e =(fe)(ge)= fe, so we have 3.4. ABSTRACT BOSETS 52 e

f fg g

(fg)e = fe ge

We can think of this as an arrow analogue of the surjectivity of right translation by e in an appropriate D-class by Green’s Lemma.

Note that for each property we have described, there is a dual property which is obtained by interchanging arrows and reversing products. It is remarkable that this small list of observations and their duals turn out to be enough to abstractly characterise systems of idempotents of semigroups in a sense made precise in the next section.

3.4. Abstract Bosets

In this section the concept of an abstract boset is introduced. Bosets were invented by Nam- booripad, initially to abstractly describe skeletons of idempotents of regular semigroups. The axioms in this section are due to Nambooripad [26, 28], and the arrow notation is due to Easdown [7].

Let E be any set with a partial multiplication over a domain DE ⊆ E × E. Define arrow relations and on E by

e f ⇐⇒ (f, e) ∈ DE and fe = e,

e f ⇐⇒ (e, f) ∈ DE and ef = e.

In practice, combinations of arrows will often arise, so it is convenient to use the notation

R = = ∩ , L = = ∩ , = ∩ .

The set E is called a boset (or biordered set) if it satisfies the following axioms.

and are reflexive and transitive, and (B1) DE = ∪ ∪ ∪ 3.4. ABSTRACT BOSETS 53

f f ⇒ (B2.1) e e ef

f f ⇒ (B2.1)* e e fe

e ⇒ fe ge (B2.2) f g

e ⇒ ef eg (B2.2)* f g

(B3.1) e f g ⇒ (eg)f = ef

(B3.1)* e f g ⇒ f(ge)= fe

e ⇒ (gf)e =(ge)(fe) (B3.2) f g

e ⇒ e(fg)=(ef)(eg) (B3.2)* f g 3.4. ABSTRACT BOSETS 54

e e

f g f f ′ g ⇒ (∃f ′ ∈ E) (B4) fe ge f ′e = fe ge

e e

f g f f ′ g ⇒ (∃f ′ ∈ E) (B4)* ef eg ef ′ = ef eg

Note that after Axiom (B1) the axioms come in dual pairs. The starred axiom can be obtained from the nonstarred axiom by interchanging the arrows and , and re- versing all of the products. Note also that these are not Nambooripad’s original axioms. Axioms (B4) and (B4)* are consequences of Nambooripad’s original list [28] Proposition 2.4, and were adopted by Easdown [4, 5] in order to use equivalent axioms which avoid sandwich sets. For a list of Nambooripad’s original axioms, using his original notation, see [17] Section VIII.3.

For any boset E the dual boset of E is the boset obtained from E by interchanging arrows and reversing all of the products, which is the boset E∗ = {e∗ | e ∈ E} whose underlying set is chosen to be in a one-one correspondence with E under the mapping e 7→ e∗ with respect to the partial operation e∗f ∗ =(fe)∗ over the domain

∗ ∗ ∗ ∗ DE∗ = {(e , f ) | (f, e) ∈ DE} = {(e , f ) | (e, f) ∈ DE}

(since DE is symmetric), so that e∗ f ∗ ⇐⇒ e f and e∗ f ∗ ⇐⇒ e f. 3.4. ABSTRACT BOSETS 55

Observe that for any statement that is true for all bosets, the dual statement, that is the statement obtained by interchanging arrows and reversing products in the original statement, is also true for all bosets. This duality, as well as a variation in Chapter 4, will be used throughout this thesis.

Observe that together the relations and possess an intertwined antisymmetric property: e f e ⇒ e = f. If the relations and coincide then E is a poset under = = . Con- versely, if E is any poset under ≤ then E becomes a boset by defining = = ≤ and ef = fe = e whenever e ≤ f.

If e f and g h then the products fe = e and gh = g are called trivial, and the products ef and hg are called nontrivial.

Axioms (B4) and (B4)* can be adjusted slightly. The proof appears in [4] and is included here as a good illustration of the arrow relations and the use of the boset axioms.

Proposition 3.3. Let E be any boset and e,f,g ∈ E. Then

(i) e e

f g f f ′ g ⇒ (∃f ′ ∈ E) fe ge f ′e = fe ge and dually (ii) e e

f g f f ′ g ⇒ (∃f ′ ∈ E) ef eg ef ′ = ef eg 3.5. SAWTOOTH BOSETS 56

Proof. By duality it is sufficient to prove (i). Suppose that e,f,g ∈ E such that e

f g

fe ge

Then by Axiom (B4) there is an element f ′ ∈ E such that e

f f ′ g

f ′e = fe ge

It remains to prove gf ′ = g, for then g f ′ and we will be done. We have gf ′ =(gf ′)g since gf ′ g = ((gf ′)e)g by Axiom (B3.1) = ((ge)(f ′e))g by Axiom (B3.2) =(ge)g since ge f ′e = g since g ge. 

The observations of the previous section verify that the set of idempotents E of any semi- group S with partial multiplication over the domain

DE = ∪ ∪ ∪ satisfies the boset axioms of this section.

3.5. Sawtooth Bosets

In this section an infinite class of examples of bosets, called sawtooth bosets, will be introduced for which = . The first examples of this class are due to Easdown and appear in [8]. Examples of this class have been studied by Jordan [23], in particular their

relationship to the general construction TE of Chapter 4. 3.5. SAWTOOTH BOSETS 57 ··· ···

··· ···

··· ···

Figure 3.12. Arrow diagram of a typical fragment of a sawtooth boset

Define a boset as follows: Let E be a disjoint union of subsets T and R, where R is an R-class, that is r s for each r, s ∈ R, and the relations and restricted to T are trivial. For each t ∈ T choose a nonempty subset ω(t) of R for which r t for all r ∈ ω(t). We will see that ω(t) = {t} ∪ ω(t) in this case. Further, add arrows r t for all r ∈ R, t ∈ T and trivial arrows e e for all e ∈ Eb. It is clear that the arrows are reflexiveb and transitive and = onb E. The nature of the arrow relations, and the trivial products they imply, on the set E = T ⊔ R are illustrated by the diagram in Figure 3.12. The R-class R forms the bottom row of the diagram and the set T forms the top row of the diagram. Arrows that can be deduced by reflexivity or transitivity are omitted. Elements of T are called teeth because of the sawtooth shape of the diagram. In order for E to become a boset we need to define a partial multiplication of E over the domain

DE = ∪ ∪ ∪ in such a way that the boset axioms are satisfied. Trivial multiplication is induced by the arrows. We must also define the nontrivial multiplication. For each tooth t and r ∈ R\ω(t) define the nontrivial product rt so that rt ∈ ω(t). Then Axiom (B1) and Axiom (B2.1) are satisfied. It remains to check the other boset axioms. Axioms (B2.1)*, (B2.2)*, (B3.1)*b and (B4)* become trivial since = . Axiomsb (B3.2) and (B3.2)* follow by a simple use of Axiom (B2.1). It remains to check Axioms (B2.2), (B3.1) and (B4). Suppose e,f,g ∈ E such that e

f g

holds. If f = g then fe = ge. Otherwise e = g ∈ T and f ∈ ω(e) so fe = f g = ge. This shows that Axiom (B2.2) is satisfied. Suppose e,f,g ∈ E such that e f g. If f = g then (eg)f =(ef)f = ef by Axiom (B2.1). Otherwise e, fb ∈ R so eg ∈ R by Axiom (B2.1) and (eg)f = f = ef. This shows that Axiom (B3.1) is satisfied. 3.5. SAWTOOTH BOSETS 58 e f

h g k

Figure 3.13. Arrow diagram of an example sawtooth boset

(i) hf = ke = g, (ii) hf = g, ke = h, (iii) hf = k, ke = g, (iv) hf = k, ke = h.

Figure 3.14. Possible nontrivial products for the arrow diagram in Figure 3.13

Suppose e,f,g ∈ E such that e

f g

fe ge

holds. If g ∈ R then ge ∈ R so fe = ge which shows Axiom (B4) is satisfied by f ′ = g. Otherwise g = e ∈ T so fe ge = g = e and (fe)e = fe which shows Axiom (B4) is satisfied by f ′ = fe.

This shows that E is a boset. Because of the underlying shape of the arrows in Figure 3.12, E is called a sawtooth boset. If |T | = n for some n ∈ Z+ then E may be called a sawtooth boset with n teeth. These bosets were studied by Jordan in [23] where he proves a criterion for the regularity of the general construction TE of Chapter 4 where E is a sawtooth boset with two teeth. Because of the underlying shape of the arrows, Jordan used the term M-boset for a sawtooth boset with 2 teeth. The first examples of sawtooth bosets appeared in [8] and are the following. Consider the diagram in Figure 3.13. Assuming transitivity of we have h f and k e. Since |ω(e)| = |ω(f)| = 2 there are four possibilities for nontrivial products which appear in Figure 3.14. Figure 3.13 is an example of an arrow diagram, and Figure 3.13 together with a validb choiceb of nontrivial products in 3.6. SUBBOSETS AND HOMOMORPHISMS 59

Figure 3.14 is an example of a boset diagram of a boset. Both diagrams are analogous to a Hasse diagram of a poset. Arrows are read transitively so that as few are included on the diagrams as possible. The arrow diagram indicates the arrow relations on the boset and the trivial multiplication implied by them. The boset diagram includes the arrow diagram and any boset products that cannot be deduced from the arrow diagram.

The bosets defined by Figure 3.13 and Figure 3.14 are the skeletons of idempotents of contrasting semigroups.

The boset with nontrivial products given in (i) comes from a five element band in which ef = fe = g.

The bosets with nontrivial products given in (ii) and (iii) come from isomorphic six element semigroups which are not regular. In (ii) the product ef equals g, and the product fe is a new element, which is not regular. In (iii) the product fe equals g, and the product ef is a new element, which is not regular.

The boset with nontrivial products given in (iv) comes from no finite semigroup. If it did come from a finite semigroup then some power (ef)n of ef would be idempotent. Then (ef)n f since (ef)nf =(ef)n and = , but (ef)n ∈{/ g,k} since g(ef)n = g and k(ef)n = k, so (ef)n = f. This implies that

ef = e(ef)n =(ef)n = f, so f e which is a contradiction.

We calculate TE later in each case.

3.6. Subbosets and Homomorphisms

A subset F of a boset E is a subboset (or biordered subset) of E if F is a boset under the partial multiplication inherited from E. This is clearly equivalent to

(i) F is closed under the partial multiplication of E, that is,

(∀e, f ∈ F ) (e, f) ∈ DE ⇒ ef ∈ F.

(ii) Axioms (B4) and (B4)* hold with respect to DF . 3.6. SUBBOSETS AND HOMOMORPHISMS 60

Let E be a boset and e ∈ E. Define

ωr(e)= {x ∈ E | x e},

ωℓ(e)= {x ∈ E | x e}, and

ω(e)= ωr(e) ∩ ωℓ(e)= {x ∈ E | x e}.

The sets ωr(e), ωℓ(e) and ω(e) are called the right ideal, left ideal and principal ideal respectively, of E generated by e.

The following result was proved by Nambooripad [28], but we give an alternative proof using axioms (B4) and (B4)*.

Lemma 3.4. Let E be any boset and d ∈ E. Then the right ideal ωr(d) and the left ideal

ωℓ(d) are both subbosets of E.

Proof. By duality, it is sufficient to prove that the right ideal ωr(d) is a subboset of E.

First, we will prove that ωr(d) is closed under the partial multiplication inherited from E.

Suppose e, f ∈ ωr(d) such that (e, f) ∈ DE. If e f then ef f d; if e f then ef = f d; if e f then ef = e d; and if e f then ef e d. In any case ef ∈ ωr(d), so ωr(d) is closed under the multiplication inherited from E. Next we prove that Axiom (B4) holds in ωr(d). Suppose e,f,g ∈ ωr(d) such that e

f g

fe ge

By Axiom (B4) (for E) there exists f ′ ∈ E such that e

f f ′ g

f ′e = fe ge

′ ′ Then f e d, so f ∈ ωr(d) which shows that Axiom (B4) holds in ωr(d). Finally, we

prove that Axiom (B4)* holds in ωr(d). Suppose that e,f,g ∈ ωr(d) such that 3.6. SUBBOSETS AND HOMOMORPHISMS 61 e

f g

ef eg

By Axiom (B4)* (for E) there exists f ′ ∈ E such that

e

f f ′ g

ef ′ = ef eg

′ ′ Then f g d, so f ∈ ωr(d) which shows that Axiom (B4)* holds in ωr(d). 

The following corollary was proved by Jordan [23], Lemma 2.7.1.

Corollary 3.5. Let E be any boset and d ∈ E. Then the principal ideal ω(d) is a subboset of E.

Proof. By Lemma 3.4, since ω(d)= ωr(d)∩ωℓ(d). The principal ideal ω(d) is closed under

the partial multiplication inherited from E since both ωr(d) and ωℓ(d) are. Any element ′ f ∈ E which satisfies Axiom (B4) or (B4)* (for ω(d)) is in ω(d) since it is in both ωr(d)

and ωℓ(d). 

A mapping φ : E → F between two bosets E and F is a boset homomorphism if φ respects the partial multiplication in E, that is,

(∀e, f ∈ E) (e, f) ∈ DE ⇒ (eφ)(fφ)=(ef)φ.

Observe that if φ preserves arrows, that is e f implies eφ fφ and e f implies eφ fφ, then φ respects the trivial products. More work may be needed to check if φ respects nontrivial products. The homomorphism φ is a boset isomorphism if it is bijective and both φ and φ−1 are boset homomorphisms. In this case E and F are isomorphic to each other and we may write E ∼= F .

Consider the following example. Let E and F be the sawtooth bosets with arrow diagram in Figure 3.13 and nontrivial products given by Figure 3.14 (ii) and (iii) respectively. We 3.6. SUBBOSETS AND HOMOMORPHISMS 62

show that E and F are isomorphic bosets, so there are at most three different bosets with arrow diagram in Figure 3.13. Consider the bijection φ : E → F defined by

e 7→ f, f 7→ e, g 7→ g, h 7→ k, k 7→ h.

Clearly φ is an arrow preserving bijection, so φ and φ−1 respect the trivial products. It remains to check that they both respect the nontrivial products. We have

(hφ)(fφ)= ke = g = gφ =(hf)φ,

(kφ)(eφ)= hf = k = hφ =(ke)φ and (hφ−1)(fφ−1)= ke = h = kφ−1 =(hf)φ−1, (kφ−1)(eφ−1)= hf = g = gφ−1 =(ke)φ−1. Hence E =∼ F .

Observe that both φ and φ−1 may be considered as the unique nontrivial automorphism of the arrow diagram in Figure 3.13. This illustrates an important idea. Given any arrow diagram for a boset, it is useful to know when two different choices for the nontrivial products produce isomorphic bosets. This can be done in the following way. Two choices of nontrivial products produce isomorphic bosets if and only if the equations defining one choice can be obtained from the equations defining the other choice by applying an automorphism of the arrow diagram. This is particularly useful for sawtooth bosets as the automorphisms of the arrow diagrams are easily identified. This allows us, given any arrow diagram for a sawtooth boset, to quickly generate a minimal list of choices of nontrivial products which is complete in the sense that for any sawtooth boset with the given arrow diagram there is precisely one set of nontrivial products in the list which defines it. Consider the arrow diagram in Figure 3.13. There are exactly three bosets with this arrow diagram. The nontrivial products are given by Figure 3.14 (i), (ii) and (iv) since applying the unique nontrivial automorphism φ fixes the sets of equations in (i) and (iv).

Let E be any boset and e, f ∈ E. A boset isomorphism α : ω(e) → ω(f) is called a principal ideal isomorphism.

If E is the boset in the previous example then there are two principal ideal isomorphisms ω(e) → ω(f). They are given by

α : e 7→ f, g 7→ g, h 7→ k,

and β : e 7→ f, g 7→ k, h 7→ g. 3.7. BOSETS ARE SKELETONS OF IDEMPOTENTS 63

It is easy to see that both α and β are arrow preserving bijections. This implies that they are both boset isomorphisms, since there are no nontrivial products in the principal ideals ω(e) and ω(f).

3.7. Bosets are Skeletons of Idempotents

Recall from Section 3.4 that the skeleton of idempotents of any semigroup is a boset. In this section, we discuss a result, due to Easdown [2], that any boset is the skeleton of idempotents of some semigroup. These results together show that bosets are precisely skeletons of idempotents of semigroups.

Theorem 3.6. Let E be any boset. Then E ∼= E(S) for some semigroup S. 

∼ The proof can be found in [2], Theorem 3.3 and involves showing that E = E(FE) where

FE is the free semigroup on the boset E, which is given by

FE = hE | (∀(e, f) ∈ DE) ef = e ⋆ fi, where ⋆ is the partial multiplication of E. The function which maps the boset element e ∈ E to the word e ∈ FE is a boset isomorphism E → E(FE).

This result shows that any boset E is the boset of idempotents of some semigroup S. It follows quickly that the dual boset E∗ is the boset of idempotents of the dual semigroup S∗ so boset duality is semigroup duality when attention is paid only to the idempotents.

This result also gives a very useful corollary.

Corollary 3.7. Let E be any boset. Then the partial multiplication in E is associative, to the extent that, if one defined product is rebracketed to give another defined product then the two products are equal. 

The proof is by Theorem 3.6, the definition of boset isomorphism, and associativity in the semigroup FE. 3.8. REGULAR BOSETS 64

3.8. Regular Bosets

In this section the concept of a regular boset is introduced, which is the skeleton of idem- potents of a regular semigroup. Regular bosets form the basis for Nambooripad’s theory of fundamental regular semigroups [26, 28]. They are defined in terms of sandwich sets.

Let E be any boset and e, f ∈ E. Define

U(e, f)= {g ∈ E | e g f} and S(e, f)= {g ∈ U(e, f) | (∀h ∈ U(e, f)) eh eg and hf gf}. The set S(e, f) is called the sandwich set of the pair (e, f) ∈ E ×E. The boset E is regular if the sandwich set S(e, f) is nonempty for all pairs (e, f) ∈ E × E.

The sandwich set of a pair of boset elements is a generalisation of the greatest lower bound of a pair of poset elements. If E is any poset under = ∩ and e, f ∈ E then U(e, f) is the set of lower bounds of e and f, and the sandwich set S(e, f) is the singleton set containing the greatest lower bound of e and f if it exists and is empty otherwise. It follows that regular posets are precisely semilattices.

The concept of a sandwich set of a pair of elements has been extended by Pastijn [29] to a sandwich set S(e1,...,en) of a sequence of elements e1,...,en ∈ E.

If E is the boset of a regular semigroup then by a result of FitzGerald [15], hEi is a regular semigroup. Let a be an inverse for e1 ··· en and form the FitzGerald idempotents

xi = ei+1 ··· enae1 ··· ei for each i =1,...,n−1. Pastijn [29] proved that the sequence (x1,...,xn−1) is an element of the sandwich set S(e1,...,en). He also proves that the sandwich set S(e1,...,en) is n nonempty for all sequences (e1,...,en) ∈ E if and only if the sandwich set S(e, f) is nonempty for all ordered pairs (e, f) ∈ E × E.

The next result is due to Nambooripad [28] Theorem 1.1 (a4) and Corollary 4.15, and characterises regular bosets as skeletons of idempotents of regular semigroups.

Theorem 3.8. Let E be any boset. Then E =∼ E(S) for some regular semigroup S if and only if E is regular. 

The proof is highly nontrivial and involves inductive groupoids. 3.8. REGULAR BOSETS 65 e f

x y

h g k k k ke hf k k kx hy

Figure 3.15. Boset diagram in which the boset in Figure 3.14 (iv) mini- mally embeds

The following criterion for the regularity of sawtooth bosets is due to Jordan [23] Lemma 4.3.2.

Lemma 3.9. Let E be any sawtooth boset. Then E is regular if and only if |ω(t)u| = 1 whenever t and u are different teeth.  b The proof involves showing that

ω(t) if |ω(t)u| =1 S(t, u)= ( ∅ otherwise b b whenever t and u are different teeth. This result shows that it is easy to construct infinitely many regular and nonregular examples of sawtooth bosets.

Case (i) of Figure 3.14 produces a regular boset since ω(e)f = ω(f)e = {g}. Case (ii) just fails to be regular since ω(f)e = {g, h}. Case (iv) also is nonregular since ω(f)e = {g, h} and ω(e)f = {g,k}, but can be embedded “minimally”b (in ab sense we will make precise later) in the boset in Figureb 3.15. b b If E is a regular boset the general construction TE of Chapter 4 is the maximum fundamen- tal regular semigroup with boset E. If E is a nonregular boset, then the boset of TE may be larger than E, but E embeds in the boset of TE, and TE contains every fundamental regular-generated semigroup with boset E (see Chapter 5). 3.8. REGULAR BOSETS 66

The regularity of TE where E is a nonregular boset is largely unknown. However, Jordan has proved a criterion for the regularity of TE for sawtooth bosets with 2 teeth (see Chapter 6). CHAPTER 4

A General Construction

In this chapter a representation φ of an arbitrary semigroup, due to Easdown and Hall [3], is introduced which is a generalisation of Munn’s fundamental representation of an inverse semigroup, though at the time it was not known whether the image was always fundamental. Munn’s result that every inverse semigroup is a coextension of a fundamental inverse semigroup with the same semilattice of idempotents is generalised to arbitrary semigroups. The general theory is checkered and due variously to Easdown [9, 3], Edwards [10] and Hall [20, 3]. This chapter attempts to put down for the first time a self-contained account in one place. Munn’s fundamental inverse semigroup TE of a semilattice E is also generalised to a semigroup which is also called TE where E is any boset. The relationship between TE and the fundamental representation φ is investigated, as well as properties of

TE for various classes of bosets.

4.1. The Fundamental Representation

Munn proved in [25] that every inverse semigroup is a coextension of a fundamental inverse semigroup with the same semilattice of idempotents. The aim of this section is to generalise this result to the class of all semigroups. It is natural to ask the question “is an arbitrary semigroup a coextension of a fundamental semigroup with the same boset of idempotents?” The answer is no, in general, as the following example, which is due to Easdown [9], shows. Let E be the boset with diagram

e f

and consider the free semigroup

2 2 FE = he, f | e = e, f = fi

on the boset E. Elements of FE can be uniquely represented by words of alternating e’s and f’s. Let a and b be two such words of length n and m respectively. The product ab can be represented by a word of alternating e’s and f’s of length n + m − 1 if the last letter of a is the same as the first letter of b, and of length n + m otherwise, so multiplying 67 4.1. THE FUNDAMENTAL REPRESENTATION 68

two words of alternating e’s and f’s of length greater than 1 together produces a word of alternating e’s and f’s which is strictly longer than both of the original words. This shows

that the only idempotents in FE are e and f, and there are no arrows between them. This

implies that the boset of idempotents of FE is isomorphic to E (which is also guaranteed by the proof in [2] of Theorem 3.6).

Let σ be a congruence on FE and a, b ∈ FE such that a σ b yet a =6 b in FE. Then eaf σ ebf and fae σ fbe, which implies that (ef)n σ (ef)m or (fe)n σ (fe)m for some n, m ∈ Z+ such that n =6 m. This in turn implies that some power (ef)kσ of k (ef)σ or (fe) σ of (fe)σ is idempotent in FE/σ. If σ induces an isomorphism between the

bosets of FE and FE/σ then (ef)kσ = eσ, (ef)kσ = fσ, (fe)kσ = eσ or (fe)kσ = fσ

which implies that

eσ fσ, fσ eσ, eσ fσ or fσ eσ

which is a contradiction, since there is no arrow between e and f in E.

However, we show that for every semigroup S with boset of idempotents E there is a congruence σ on S such that S/σ is fundamental and E may be regarded as a subboset of E(S/σ). More precisely, the restriction

ν|E : E → Eσ, e 7→ eσ of the natural semigroup homomorphism

ν : S → S/σ, a 7→ aσ

is a boset isomorphism. In this case we say that S is a biorder-preserving coextension of S/σ by σ, and that σ is a biorder-preserving congruence. Equivalently, σ is biorder-preserving if

e f ⇐⇒ eσ fσ, e f ⇐⇒ eσ fσ,

for all e, f ∈ E. This condition guarantees that the restriction ν|E is a bijection. Both ν|E −1 and (ν|E) are boset homomorphisms since ν respects the semigroup multiplication.

Consider again the free semigroup FE on the boset E given by 4.1. THE FUNDAMENTAL REPRESENTATION 69

e f.

There is a semigroup homomorphism from FE to the fundamental semilattice

e f

z which is given by e 7→ e, f 7→ f, a 7→ z for all a ∈ FE \{e, f}, so FE is a biorder-preserving coextension of a fundamental semigroup.

Let S be any semigroup with boset of idempotents E. Denote the R-class (resp. L -class) of E which contains e ∈ E by Re (resp. Le). Throughout this section, and whenever the homomorphism φ (defined below) appears, denote Green’s relation R (resp. L ) on the semigroup S by R◦ (resp. L ◦), and the R◦-class (resp. L ◦-class) of S which contains ◦ ◦ a ∈ S by Ra (resp. La). For each a ∈ S two partial transformations will be defined, on the sets E/L and E/R of L -classes and R-classes of E. For convenience, the element ∞, which is not in E/L or E/R, will be introduced, to disguise the partial transformations as full transformations. Let a ∈ S and define

ρa ∈ TE/L ∪{∞}

◦ ◦ Lxa ∩ E if x R xa Lx 7→ ( ∞ otherwise ∞ 7→ ∞ and

λa ∈ TE/R∪{∞}

◦ ◦ Rax ∩ E if x L ax Rx 7→ ( ∞ otherwise ∞ 7→ ∞.

Observe that λa is the dual of ρa. The definition for λa can be obtained from the definition ◦ ◦ for ρa by interchanging L -classes with R-classes, L -classes with R -classes, and reversing 4.1. THE FUNDAMENTAL REPRESENTATION 70

all of the products. Note also that ρa and λa are well defined, by Green’s Lemma and since every regular L ◦-class and R◦-class of S contains an idempotent.

∗ Put φa =(ρa,λa) and define a mapping

∗ φ : S → TE/L ∪{∞} × TE/R∪{∞}, a 7→ φa.

Proposition 4.1. Let S be any semigroup. Then φ is a semigroup homomorphism.

Proof. Let a, b ∈ S. It is sufficient to show that ρaρb = ρab, for then by duality λbλa = λab, ◦ whence φaφb = φab. Let x ∈ E. Suppose first Lxρaρb =6 ∞. Then Lxρa =6 ∞, so x R xa ◦ ◦ and Lxρa = Lxa ∩ E = Ly for some y ∈ E. Now Lyρb = Lxρaρb =6 ∞, so y R yb and ◦ Lyρb = Lyb ∩ E. By Green’s Lemma we have the following egg-box diagram.

y ··· yb . . . . x ··· xa ··· xab

Hence we have

◦ ◦ Lxρaρb = Lyb ∩ E = Lxab ∩ E = Lxρab.

◦ ◦ Conversely, suppose now Lxρab =6 ∞. Then x R xab and Lxρab = Lxab ∩ E, so there exists 1 ◦ ◦ s ∈ S such that xabs = x, which implies that x R xa, so Lxρa = Lxa ∩ E. Consider the FitzGerald idempotent bsxa ∈ E. It is easy to check that bsxa L ◦ xa and bsxa R◦ bsxab. So we have

◦ ◦ ◦ Lxρaρb =(Lxa ∩ E)ρb = Lbsxaρb = Lbsxab ∩ E = Lxab ∩ E = Lxρab.

This shows that ρaρb = ρab which completes the proof. 

This representation is a modification of a representation φ◦, due to Easdown and Hall [3], of S by transformations and dual transformations of the sets X of regular L ◦-classes, and Y of regular R◦-classes respectively of S. This representation is defined by

◦ ∗ ◦ ◦ ◦ ∗ φ : S → TX∪{∞} × TY ∪{∞}, a 7→ φa =(ρa, (λa) ) 4.1. THE FUNDAMENTAL REPRESENTATION 71

where

◦ ρa ∈ TX∪{∞}

◦ ◦ ◦ Lxa if x R xa Lx 7→ ( ∞ otherwise ∞ 7→ ∞

and

◦ λa ∈ TY ∪{∞}

◦ ◦ ◦ Rax if x L ax Rx 7→ ( ∞ otherwise ∞ 7→ ∞.

The two representations φ and φ◦ are equivalent since the L ◦-class and R◦-class of an idempotent are both regular, and every regular L ◦-class and R◦-class contains an idem- potent. The representation φ◦ first appeared in [3] and [10], but similar representations have been used earlier. See for example [20] or [18]. The proofs in this section use the representation φ, which is new.

Put µ = µ(S) = ker φ. It follows quickly from the definition of φ that

µ = (a, b) ∈ S × S (∀x ∈ E) each of x R◦ xa, x R◦ xb implies xa H xb  and each of x L ◦ ax, x L ◦ bx implies ax H bx .

The congruence µ has been studied extensively by Edwards [10, 11, 12, 13, 14]. He proves the following theorem, using the representation φ◦, with two extra equivalent conditions, one of which involves the concept of an eventually regular semigroup, which can be found in [10] Theorem 9.

Theorem 4.2. Let S be any semigroup and σ any congruence on S. Then σ ⊆ µ if and only if σ is a H -congruence.

Proof. Suppose first that σ ⊆ µ. Let e σ a for e ∈ E and a ∈ S. Then φe = φa since σ ⊆ µ. This implies that ◦ Le = Leρe = Leρa = Lea ∩ E, and ◦ Re = Reλe = Reλa = Rae ∩ E, 4.1. THE FUNDAMENTAL REPRESENTATION 72

so ◦ ◦ ◦ ◦ ◦ ◦ Le = Lea ≤ La and Re = Rae ≤ Ra, giving He ≤ Ha which proves σ is an H -congruence.

Conversely, suppose now that σ is an H -congruence. Let a σ b for a, b ∈ S. We show that

ρa = ρb, for then λa = λb by duality, so that φa = φb, that is a µ b, and we are done. Let ◦ x ∈ E. Suppose first that Lxρa =6 ∞, so x R xa. Hence xa is regular, so choose an inverse (xa)′ for xa. Then (xa)′xb σ (xa)′xa ∈ E, so H(xa)′xa ≤ H(xa)′xb, since σ is an H -congruence, whence

◦ ◦ ◦ ◦ Lxa = L(xa)′xa ≤ L(xa)′xb ≤ Lxb. Also xb(xa)′ σ xa(xa)′ ∈ E, so Hxa(xa)′ ≤ Hxb(xa)′ , again since σ is an H -congruence, whence

◦ ◦ ◦ ◦ ◦ ◦ Rx = Rxa = Rxa(xa)′ ≤ Rxb(xa)′ ≤ Rxb ≤ Rx,

◦ ◦ ◦ ◦ ◦ so that x R xb. Similarly Lxb ≤ Lxa, so Lxb = Lxa, yielding

◦ ◦ Lxρs = Lxa ∩ E = Lxb ∩ E = Lxρb.

By symmetry Lxρb =6 ∞ implies Lxρb = Lxρa, so that ρa = ρb. 

The following corollary follows immediately from Theorem 4.2.

Corollary 4.3. Let S be any semigroup. Then µ is the maximum H -congruence on S.

Hall proves in [20] Theorem 5, that µ is the maximum congruence contained in H on a regular semigroup. It was also well-known that µ is the maximum idempotent separating congruence on a regular semigroup. Proofs appear in [3] Theorem 1, and also in [10] Theorem 9 (for eventually regular semigroups). Both results are included here as a corollary to Theorem 4.2.

Corollary 4.4. Let S be any regular semigroup. Then µ is the maximum congruence contained in H , and also the maximum idempotent separating congruence on S.

Proof. Let σ be any congruence on a regular semigroup S. It is sufficient to prove that

σ ⊆ H ⇐⇒ σ is an H -congruence ⇐⇒ σ is idempotent separating, 4.1. THE FUNDAMENTAL REPRESENTATION 73

for then by Theorem 4.2, the congruence µ is the maximum congruence contained in H , and also the maximum idempotent separating congruence on S.

We first prove that σ ⊆ H implies σ is a H -congruence. Suppose σ ⊆ H . Let e ∈ E and a ∈ S such that e σ a. Then e H a since σ ⊆ H , so σ is a H -congruence.

The claim that σ is a H -congruence implies σ is idempotent separating is immediate, since this is true for congruences on arbitrary semigroups.

Finally we prove that σ is idempotent separating implies σ ⊆ H . Suppose σ is idempotent separating. Let a, b ∈ S such that a σ b. By symmetry and duality it is sufficient to prove ◦ ◦ ′ ′ ′ Ra ≤ Rb , for then a H b, so σ ⊆ H . Let a ∈ S be an inverse for a. Then aa σ ba . ′ By Lallement’s Lemma there is an idempotent f ∈ E such that f σ ba and Hf ≤ Hba′ . ′ This implies, since σ is idempotent separating, that aa = f and Haa′ ≤ Hba′ . Hence ◦ ◦ ◦ ◦ Ra = Raa′ ≤ Rba′ ≤ Rb . 

The following example shows that this is not true for an arbitrary semigroup. Consider again the free semigroup 2 2 FE = he, f | e = e, f = fi on the boset E given by

e f.

Recall that multiplying two words of alternating e’s and f’s of length greater than 1 together produces a word of alternating e’s and f’s which is strictly longer than both of ◦ ◦ the original words. This implies that R and L are trivial, so we may identify a ∈ FE ◦ ◦ with Ra = La = Ha = {a}.

Recall from Section 2.5 that FE has no maximum idempotent separating congruence. We

now show that µ is nontrivial, so is not contained in H . We will show that φef = φfe,

for then ef µ fe, so µ is nontrivial. By duality it is enough to show ρef = ρfe, for then

λef = λfe and hence φef = φfe. For each a ∈ S we have ea if e = ea ρa : e 7→ ( ∞ otherwise fa if f = fa f 7→ ( ∞ otherwise. ∞ 7→ ∞

It follows quickly that ρa = ρb for all a, b ∈ S \{e, f}. In particular ρef = ρfe. 4.1. THE FUNDAMENTAL REPRESENTATION 74

The next result is due to Easdown and Hall, and appears in [3] Theorem 6.

Theorem 4.5. Let S be any semigroup. Then µ is a biorder-preserving congruence on S.

Proof. Let e, f ∈ E. By duality it is sufficient to prove e f if and only if eµ fµ. Suppose e f. Then (fµ)(eµ)=(fe)µ = eµ, so eµ fµ. Conversely, suppose eµ fµ. Then

(fe)µ =(fµ)(eµ)= eµ, so φe = φfe. In particular λe = λfe, so

◦ ◦ Re = Reλe = Reλfe = R((fe)e) ∩ E = Rfe ∩ E and e L ◦ (fe)e = fe.

Also, there is an element s ∈ S1 such that fe = es, so

(fe)(fe)= fees = fes = ffe = fe.

This shows that e and fe are H -equivalent idempotents, so e = fe. That is, e f. 

The next result is due to Edwards and appears in [11] Theorem 8.

Theorem 4.6. Let S be any semigroup. Then µ (S/µ)=1.

Proof. Denote the congruence µ on S/µ by µ′. Let a, b ∈ S such that (aµ, bµ) ∈ µ′. It is ′ sufficient to prove ρa = ρb, for then by duality λa = λb, so φa = φb, that is a µ b, so µ is the trivial congruence on S/µ and we will be done. Let x ∈ E such that Lxρa =6 ∞. Then x R◦ xa. Because µ is a congruence,

xµ ∈ E(S/µ) and xµ R◦ (xa)µ =(xµ)(aµ).

Hence

◦ ◦ L(xa)µ ∩ E(S/µ)= Lxµρaµ = Lxµρbµ = L(xb)µ ∩ E(S/µ) and (xa)µ R◦ xµ R◦ (xb)µ, 4.1. THE FUNDAMENTAL REPRESENTATION 75

so there exists s,t,u ∈ S1 such that (xa)µ =(xb)µ(sµ)=(xbs)µ, (xa)µ =(tµ)(xb)µ =(txb)µ, (xb)µ =(uµ)(xa)µ =(uxa)µ. In particular,

ρxa = ρxbs = ρtxb and ρxb = ρuxa. But x R◦ xa = x(xa), so

◦ ◦ Lxa ∩ E = Lxρxa = Lxρxbs = Lxρtxb = Lxtxb ∩ E and x R◦ xtxb R◦ xxbs = xbs. Thus ◦ ◦ ◦ ◦ Rx = Rxbs ≤ Rxb ≤ Rx, ◦ ◦ ◦ so Rxa = Rx = Rxb and ◦ ◦ Lxa = Lxtxb ≤ Lxb.

Since ρxb = ρuxa, in particular

◦ ◦ Lxb ∩ E = Lxρxb = Lxρuxa = Lxuxa ∩ E so that ◦ ◦ ◦ Lxb = Lxuxa ≤ Lxa. Thus ◦ ◦ Lxρa = Lxa ∩ E = Lxb ∩ E = Lxρb.

By symmetry, this is enough to prove that ρa = ρb. 

The next result is due to Easdown and appears in [9] Theorem 4.

Theorem 4.7. Let S be any semigroup. Then S is fundamental if and only if µ =1.

Proof. Suppose first S is fundamental. Then µ = 1, since µ is idempotent separating. Conversely, now suppose µ = 1. Let σ be any idempotent separating congruence on S. We show that σ is an H -congruence for then σ ⊆ µ by Theorem 4.2, so σ = 1 and we will be done.

Let e ∈ E, a ∈ S such that e σ a. We show that He ≤ Ha. It is sufficient to show that φa3 is idempotent, for then a3 is idempotent since φ is a semigroup isomorphism. So e = a3, 3 2 2 since e σ a and σ is idempotent separating. Then e = a(a )=(a )a, so He ≤ Ha and we will be done. 4.1. THE FUNDAMENTAL REPRESENTATION 76

By duality it is sufficient to show that ρa3 is idempotent. Let L be any L -class. If

Lρa3 = ∞ then Lρa3 = Lρa3 ρa3 . Suppose that Lρa3 =6 ∞ then L = Lx for some x ∈ E and x R◦ xa3. Then there exists s ∈ S1 such that x = xa3s. We have the following egg-box diagram.

x = xa3s ··· xa ··· xa2 ...... asx ··· asxa ··· asxa2 ...... a2sx ··· a2sxa ··· a2sxa2

The elements a2sxa and asxa2 are both FitzGerald idempotents. We also have a σ e = e2 σ a2, so a2sxa σ asxa2. This implies that a2sxa = asxa2, since σ is idempotent separat- ing. This in turn implies that xa H xa2, so xa H xan for all n ∈ Z+, by Green’s Lemma. In particular xa3 L ◦ xa6, so

◦ ◦ Lρa3 = Lxa3 ∩ E = Lxa6 ∩ E = Lρa6 = Lρa3 ρa3 .

This shows that ρa3 is idempotent. 

The next result generalises Munn’s result that every inverse semigroup is a coextension of a fundamental inverse semigroup with the same semilattice of idempotents to the class of all semigroups. It is due to Easdown and appears in [9] Corollary 5. The proof follows immediately by Theorem 4.5, Theorem 4.6 and Theorem 4.7.

Corollary 4.8. Let S be any semigroup with boset of idempotents E. Then S/µ is funda- mental and E is isomorphic to a subboset of E(S/µ). Thus every semigroup is a biorder- preserving coextension of a fundamental semigroup. 

We finish this section by describing what happens when the representation φ is applied to an example. Applying φ to the nonregular semigroup End(D8) (see Section 3.2, Figure 3.11) yields the following calculation. First we label the nontrivial L -classes and R- classes of E = E(End(D8)) in Figure 4.1. Elements of image of φ appear in Figure 4.2. ′ ′ ′ ′ ′ ′ ′ The notation ( L1 L2 L3 L4 | R1 R2 R3 ) used for a typical element φθ means that

′ ρθ : Li 7→ Li for i =1, 2, 3, 4, and ′ λθ : Rj 7→ Rj for j =1, 2, 3. 4.2. THE CONSTRUCTION OF TE 77

[ab, 1] [a3b, 1] R3

[b, b] [a2b, a2b] R2

R1 [1, b] [1, ab] [1, a2b] [1, a3b]

L1 L2 L3 L4

Figure 4.1. Nontrivial L -classes and R-classes of E(End(D8))

Thus all but one of the nonregular D-classes of End(D8) merge with the regular D-classes to give the egg-box diagram in Figure 4.3 for the fundamental semigroup [End(D8)]φ, with just one nonregular D-class.

4.2. The Construction of TE

In this section a construction is introduced, which is a natural generalisation of the Munn semigroup and appears for the first time in [23].

Let E be any boset. For each principal ideal isomorphism α : ω(e) → ω(eα) of E, two partial transformations will be defined, on the sets E/L and E/R of L -classes and R- classes of E. Again, the element ∞, which is not in E/L or E/R, will be introduced to disguise partial transformations as full transformations. Let α : ω(e) → ω(eα) be a principal ideal isomorphism of E and define

ρα ∈ T (E/L ∪ {∞}) L if x e for some x ∈ L L 7→ (xe)α ( ∞ otherwise ∞ 7→ ∞ 4.2. THE CONSTRUCTION OF TE 78

φ1 = ( L1 L2 L3 L4 | R1 R2 R3 )

φα = ( L2 L3 L4 L1 | R1 R3 R2 ) 2 φα = ( L3 L4 L1 L2 | R1 R2 R3 ) 3 φα = ( L4 L1 L2 L3 | R1 R3 R2 ) φβ = ( L1 L4 L3 L2 | R1 R2 R3 )

φαβ = ( L4 L3 L2 L1 | R1 R3 R2 )

φα2β = ( L3 L2 L1 L4 | R1 R2 R3 )

φα3β = ( L2 L1 L4 L3 | R1 R3 R2 )

φγ = ( L1 L3 L1 L3 | R1 R1 − )

φγα = ( L2 L4 L2 L4 | R1 − R1 )

φγα2 = ( L3 L1 L3 L1 | R1 R1 − )

φγα3 = ( L4 L2 L4 L2 | R1 − R1 )

φδ = ( − L1 − L1 | R3 R3 − )

φδα = ( − L2 − L2 | R3 − R3 )

φδα2 = ( − L3 − L3 | R3 R3 − )

φδα3 = ( − L4 − L4 | R3 − R3 )

φαδ = ( L1 − L1 − | R2 R2 − )

φαδα = ( L2 − L2 − | R2 − R2 )

φαδα2 = ( L3 − L3 − | R2 R2 − )

φαδα3 = ( L4 − L4 − | R2 − R2 )

φǫ = ( − L1 − L1 | R3 R3 − )

φǫα = ( − L2 − L2 | R3 − R3 )

φǫα2 = ( − L3 − L3 | R3 R3 − )

φǫα3 = ( − L4 − L4 | R3 − R3 )

φαǫ = ( L1 − L1 − | R2 R2 − )

φαǫα = ( L2 − L2 − | R2 − R2 )

φαǫα2 = ( L3 − L3 − | R2 R2 − )

φαǫα3 = ( L4 − L4 − | R2 − R2 )

φχ = ( L1 L1 L1 L1 | R1 R1 − )

φχα = ( L2 L2 L2 L2 | R1 − R1 )

φχα2 = ( L3 L3 L3 L3 | R1 R1 − )

φχα3 = ( L4 L4 L4 L4 | R1 − R1 )

φ[1,a2] = ( −−−−|−−− )

φ[a2,a2] = ( −−−−|−−− )

φ[a2,1] = ( −−−−|−−− )

φ[1,1] = ( −−−−|−−− )

Figure 4.2. Elements of (End(D8))φ 4.2. THE CONSTRUCTION OF TE 79

φ1 φα φα2 φα3 ∼ = D8 φβ φαβ φα2β φα3β

φγ φγα

φγα2 φγα3

φǫα φǫα2 φǫα3 φǫ

φαǫ φαǫα φαǫα2 φαǫα3

∼ = C1 φχ φχα φχα2 φχα3

∼ = C1 φ[1,1]

Figure 4.3. Egg-box diagram of (End(D8))φ 4.3. AN IMPORTANT CLASS OF PRINCIPAL IDEAL ISOMORPHISMS 80

and

λα ∈ T (E/R ∪ {∞})

R −1 if x eα for some x ∈ R R 7→ ((eα)x)α ( ∞ otherwise ∞ 7→ ∞.

Note also that ρα and λα are well defined by Axioms (B2.2) and (B2.2)*. Put

∗ ∗ φα =(ρα,λα) ∈ TE/L ∪{∞} × TE/R∪{∞}.

The element φα originally appears in [3], in the case where α is an identity mapping 1ω(e) on a principal ideal ω(e).

Define a subsemigroup,

TE = φα α is a principal ideal isomorphism of E ,

∗ of TE/L ∪{∞} ×TE/R∪{∞}. Observe that λα can be obtained from ρα by interchanging arrows, reversing products, and interchanging L with R, e with eα, and α with α−1. This leads to a variation of boset duality:

For any statement about TE which is true for all bosets, the dual statement, that is the statement obtained by interchanging arrows, reversing products and interchanging L - classes with R-classes, principal ideal isomorphisms with their inverses, and ρ’s with λ’s in the original statement, is also true for all bosets. This duality will be used throughout this thesis.

The semigroup TE has been studied by Jordan [23], who reproves results of Hall [20], Grillet [18, 19], Nambooripad [26, 28] and Clifford [1], for fundamental regular semigroups. He

also investigates TE for some sawtooth bosets, which were introduced in Section 3.5, many of which are nonregular. To this day, Jordan’s work remains unpublished. For completeness and because of the unavailability of Jordan’s thesis [23], many of his results are discussed

here, so that this thesis is a self-contained account of the general construction TE.

4.3. An Important Class of Principal Ideal Isomorphisms

In this section an important class of principal ideal isomorphisms will be introduced, which

will be used to demonstrate the relationship between the semigroup TE and the represen- tation φ. These isomorphisms were used by Jordan [23] and are a generalisation of the 4.3. AN IMPORTANT CLASS OF PRINCIPAL IDEAL ISOMORPHISMS 81

principal ideal isomorphisms of the semilattice of idempotents of an inverse semigroup S, which were used by Munn [25], and are defined by

ω(aa−1) → ω(a−1a), x 7→ a−1xa for each a ∈ S. Let S be any semigroup, a ∈ Reg(S) and a′ ∈ V (a). Define a mapping

′ ′ ′ θa,a′ : ω(aa ) → ω(a a), x 7→ a xa.

This mapping is the restriction to the boset of S of a D-class preserving (semigroup) isomorphism used by Hall in [20] Result 3. Restricting the multiplication in Hall’s proof to the domain of the boset multiplication shows that θa,a′ is a principal ideal (boset) isomorphism. See also [22] Therorem 5.4.4 for the principal ideal (semilattice) isomorphism used earlier by Munn. The following result shows the relationship between some elements of the image of the fundamental representation φ and some elements of the generating set of the general construction TE.

′ Proposition 4.9. Let S be any semigroup, a ∈ Reg(S) and a ∈ V (a). Then φθa,a′ = φa.

Proof. By duality it is sufficient to prove ρθa,a′ = ρa. Suppose first Lρa =6 ∞. Then there ◦ ◦ ′ ◦ 1 exists x ∈ L such that x R xa R xaa and Lρa = Lxa ∩ E, so there exists s ∈ S such that x = xaa′s. Consider the FitzGerald idempotent aa′sx ∈ E. It is easy to check that xa L ◦ a′sxa, aa′sx ∈ L and aa′sx aa′, so we have

′ ′ ◦ Lρθa,a′ = Laa sxρθa,a′ = La sxa = Lxa ∩ E = Lρa.

Conversely, suppose now Lρθa,a′ =6 ∞. Then there exists x ∈ L such that

aa′ a′a

x xaa′ a′xa

′ ◦ ′ ◦ and Lρθa,a′ = La xa. Now x R xaa R xa so

◦ ′ Lρa = Lxa ∩ E = La xa = Lρθa,a′ .

Hence ρθa,a′ = ρa. 

′ This result shows that φθa,a′ is independent of choice of inverse a of a. 4.4. PRODUCTS IN TE 82

4.4. Products in TE

In this section we present some results which are useful for manipulating products in TE. Many of these results and their proofs are due to Jordan and appear in [23].

The first result, the proof of which is straightforward, appears in [23] Proposition 3.3.2.

Proposition 4.10. Let E be any boset and α : ω(e) → ω(eα) and β : ω(eα) → ω((eα)β) be principal ideal isomorphisms of E. Then φαφβ = φαβ. 

The next two results follow immediately from Proposition 4.10 and appear in [23] Corollary 3.3.3 and Corollary 3.3.6.

Corollary 4.11. Let E be any boset and α : ω(e) → ω(eα) any principal ideal isomorphism

of E. Then φeφα = φα = φαφeα. 

Corollary 4.12. Let E be any boset and α any principal ideal isomorphism of E. Then

φα is a regular element of TE.

Proof. φαφα−1 φα = φαα−1α = φα. 

The next result is due to Jordan and appears in [23] Proposition 3.3.7. The proof is included for completeness and as a good demonstration of the use of arrow diagrams associated with the definition of TE.

Lemma 4.13. Let α : ω(e) → ω(eα) be a principal ideal isomorphism of a boset E and g ∈ E such that g e. Then φgφα = φαφgα.

Proof. By duality it is sufficient to prove ρgρα = ραρgα. If Lρgρα =6 ∞ then there exists x ∈ L such that

e eα

g gα

x xg xe (xg)α (xe)α 4.4. PRODUCTS IN TE 83 and

Lραρgα = L(xe)αρgα = L((xe)α)(gα) = L((xe)g)α = L(xg)α = Lρgρα where the fourth equality is by Axiom (B3.1).

Conversely suppose Lραρgα =6 ∞. Then there exists x ∈ L, y ∈ L(xe)α such that

e eα

g gα

x xe yα−1 (xe)α y

and by Proposition 3.3 (i) there exists z such that

e

yα−1 z x

ze = yα−1 xe.

Note that z g by transitivity of . So we have

Lρgρα = Lzρgρα = Lzgρα = L(zg)α = L((ze)g)α = L((yα−1)g)α = Ly(gα) = L(xe)αρgα = Lραρgα where the fourth equality is by Axiom (B3.1). Hence ρgρα = ραρgα. 

The following two results are useful for proving that TE is fundamental for any boset E (see Chapter 5).

Lemma 4.14. Let E be any boset, α : ω(e) → ω(eα) any principal ideal isomorphism of E and x, y ∈ E such that x y e. Then the mapping

γ : ω(x) → ω((ye)α), z 7→ [(yz)e]α is a principal ideal isomorphism and φγ = φxφα.

Proof. We first show that γ is a well defined boset homomorphism. Let u ∈ ω(x). By transitivity of arrows and Axioms (B2.1) and (B2.1)* we have 4.4. PRODUCTS IN TE 84 e

x y ye (ye)α = xγ

u yu (yu)(ye)=(yu)e ((yu)e)α = uγ

where the equality (yu)(ye)=(yu)e is by Corollary 3.7. This shows γ is a well defined mapping. Let u, v ∈ ω(x). Suppose first u v. This implies that yu yv by Axiom (B2.2)* and transitivity of , which in turn implies that (yu)e (yv)e by Axiom (B2.1) and transitivity of . Applying α gives uγ = ((yu) e) α ((yv) e) α = vγ, so we have

(vγ)(uγ)= uγ =(vu)γ and

(uγ)(vγ) = (((yu)e)α)(((yv)e)α) = (((yu)e)((yv)e))α since α is a homomorphism = (((yu)(e(yv)))e)α by Corollary 3.7 = (((yu)(yv))e)α since yv e = ((y(uv))e)α by Axiom (B3.2)* =(uv)γ.

Suppose now u v. This implies implies that yu yv by Axiom (B2.1)* and transitivity of , which in turn, implies that (yu)e (yv)e by Axiom (B2.2) and transitivity of . Applying α gives uγ = ((yu)e)α ((yv)e)α = vγ, so we have

(uγ)(vγ)= uγ =(uv)γ

and

(vγ)(uγ) = (((yv)e)α)(((yu)e)α) = (((yv)e)((yu)e))α since α is a homomorphism = (((yv)(yu))e)α by Axiom (B3.2) = ((y((vy)u))e)α by Corollary 3.7 = ((y(vu))e)α since v y =(vu)γ.

This shows that γ is a homomorphism. Define a mapping

γ−1 : ω((ye)α) → ω(x), z 7→ x((zα−1)y). 4.4. PRODUCTS IN TE 85

−1 −1 −1 We will show that γ is a well defined homomorphism, γγ =1ω(x) and γ γ =1ω((ye)α), which proves that γ and γ−1 are mutually inverse principal ideal isomorphisms, justifying the notation. Let u ∈ ω((ye)α). By Axioms (B2.1) and (B2.1)* we have

(ye)α ye y x = ((ye)α)γ−1

u uα−1 (uα−1)y x((uα−1)y)= uγ−1.

This shows that γ−1 is a well defined mapping. Let u, v ∈ ω((ye)α). Suppose first u v. Applying α−1 gives uα−1 vα−1, which implies that (uα−1)y (vα−1)y by Axiom (B2.1) and transitivity of , which in turn implies that uγ−1 = x((uα−1)y) x((vα−1)y) = vγ−1 by Axiom (B2.2)* and transitivity of , so we have

(vγ−1)(uγ−1)= uγ−1 =(vu)γ−1

and

(uγ−1)(vγ−1)=(x((uα−1)y))(x((vα−1)y)) = x(((uα−1)y)((vα−1)y)) by Axiom (B3.2)* = x(((uα−1)(y(vα−1)))y) by Corollary 3.7 = x(((uα−1)(vα−1))y) since vα−1 y = x(((uv)α−1)y) since α−1 is a homomorphism =(uv)γ−1.

Suppose now u v. Applying α−1 gives uα−1 vα−1, so by Axiom (B2.2) and tran- sitivity of we must have (uα−1)y (vα−1)y, which in turn implies that uγ−1 = x((uα−1)y) x((vα−1)y)= vγ−1 by Axiom (B2.1)* and transitivity of , so we have

(uγ−1)(vγ−1)= uγ−1 =(uv)γ−1

and

(vγ−1)(uγ−1)=(x((vα−1)y))(x((uα−1)y)) = x((((vα−1)y)x)((uα−1)y)) byCorollary3.7 = x(((vα−1)y)((uα−1)y)) since (vα−1)y x = x(((vα−1)(uα−1))y) by Axiom (B3.2) = x(((vu)α−1)y) since α−1 is a homomorphism =(vu)γ−1. 4.4. PRODUCTS IN TE 86

This shows that γ−1 is a homomorphism. Suppose that u ∈ ω(x), v ∈ ω((ye)α). Then

u(γγ−1)= x(((((yu)e)α)α−1)y) = x(((yu)e)y) = x((yu)(ey)) byCorollary3.7 = x((yu)y) since y e = x(yu) since yu y =(xy)u by Corollary 3.7 = xu since x y = u since u x

and

v(γ−1γ)=((y(x((vα−1)y)))e)α = (((yx)(vα−1))(ye))α by Corollary 3.7 = ((y(vα−1))(ye))α since x y = ((vα−1)(ye))α since vα−1 y =(vα−1)α since vα−1 ye = v.

−1 −1 −1 This shows that γγ = 1ω(x) and γ γ = 1ω((ye)α), so γ and γ are mutually inverse principal ideal isomorphisms. It remains to show that φγ = φxφα.

Suppose Lρxρα =6 ∞. Then, by transitivity of arrows and Axioms (B2.1) and (B2.1)* we have u ∈ L such that e eα

x y ye (ye)α

u ux y(ux) (y(ux))e ((y(ux))e)α =(ux)γ.

Then

Lρxρα = Luxρα = Ly(ux)ρα = L((y(ux))e)α = L(ux)γ = Lργ .

If Lρxρα = ∞ then, tracing arrows and the definitions, we see that there does not exist u ∈ L with u x, so Lργ = ∞. This shows that ρxρα = ργ. 4.4. PRODUCTS IN TE 87

Suppose that Rλγ =6 ∞. Then, again by transitivity of arrows and the axioms, we have u ∈ R such that e eα

x y ye (ye)α

((ye)α)u u

((eα)u)α−1 (eα)u.

By Axiom (B4) we have v such that e

((eα)u)α−1 v y x

ve = ((eα)u)α−1 ye

so

Rλαλx = R((eα)u)α−1 λx = Rvλx = Rxv and

Rλγ = R(((ye)α)u)γ−1 = Rx(((((ye)α)u)α−1 )y). Now,

x(((((ye)α)u)α−1)y)= x(((((ye)α)((eα)u))α−1)y) by Axiom (B3.1)* = x(((ye)((eα)u)α−1)y) since α−1 is a homomorphism = x(((ye)(ve))y) = x(((yv)e)y) by Axiom (B3.2) = x((yv)y) by Axiom (B3.1) = x(yv) since yv y = xv by Axiom (B3.1)*

so

(4.15) Rλαλx = Rλγ . 4.4. PRODUCTS IN TE 88

Suppose now that Rλαλx =6 ∞. Then, successively tracing the definitions of λα and λx, we have u ∈ R, v ∈ R((eα)u)α−1 such that e eα

x y ye (ye)α

xv v ((eα)u)α−1 (eα)u u

ve (ve)α.

We also have v y so that ve ye by Axiom (B2.2). Applying α gives (ve)α (ye)α. By transitivity of we have ve ((eα)u)α−1 so applying α gives (ve)α (eα)u. By Proposition 3.3 (ii) we have t such that eα

(ye)α (ve)α t u

(ve)α (eα)u

so t ∈ R and, by transitivity of , t (ye)α. Hence Rλγ = Rtλγ =6 ∞, so Rλγ = Rλαλx

by (4.15). This shows that λγ = λαλx, whence

∗ ∗ ∗ ∗ ∗ φγ =(ργ ,λγ)=(ρxρα,λxλα)=(ρx,λx)(ρα,λα)= φxφα. 

Lemma 4.16. Let E be any boset, αi : ω(ei) → ω(eiαi) any principal ideal on E for

i =1,...,n and x0,...,xn ∈ E such that

e1 e1α1 e2 e2α2

x0 x1 x1e1 (x1e1)α1 x2 x2e2 (x2e2)α2 ···

en enαn

··· xn xnen (xnen)αn 4.5. IDEMPOTENTS IN TE 89

Then there is a principal ideal isomorphism

γ : ω(x0) → ω((xnen)αn)

such that φγ = φx0 φα1 ··· φαn .

Proof. By induction on n. If n = 1 then the result holds by Lemma 4.14. Suppose m > 1 and, as an inductive hypothesis, the result holds for n = m − 1. By the inductive ′ hypothesis there is a principal ideal isomorphism γ : ω(x0) → ω((xm−1em−1)αm−1) such

′ that φγ = φx0 φα1 ··· φαm−1 . So we have

′ ′ ′ ′′ ′ ′′ φx0 φα1 ··· φαm−1 φαm = φγ φαm = φγ φ(xm−1em−1)αm−1 φαm = φγ φγ = φγ γ = φγ where the third equality is by Lemma 4.14 for some principal ideal isomorphism

′′ γ : ω((xm−1em−1)αm−1) → ω((xmem)αm),

and the fifth equality is since γ = γ′γ′′. The result follows by induction. 

4.5. Idempotents in TE

In this section we introduce an important boset isomorphism. We also introduce an idem-

potent generated subsemigroup SE of TE, and discuss some results involving idempotents

in TE.

Let E be any boset and S any semigroup with boset of idempotents E. For each e ∈ E we

have θe,e =1ω(e), so by Proposition 4.9, φe = φθe,e = φ1ω(e) is an idempotent in TE. Define an idempotent generated subsemigroup of TE by

SE = hφe | e ∈ Ei.

The following result is due to Easdown and its proof appears in [5] Theorem 2.

Theorem 4.17. Let E be any boset. Then the mapping

φ : E → Eφ, e 7→ φe is a boset isomorphism. 

An alternative proof is the following. By Theorem 3.6, E is the boset of idempotents of some semigroup S. By Proposition 4.9 the mapping e 7→ φe is the restriction of Easdown and Hall’s representation φ on the semigroup S to the boset E. This restriction is a boset isomorphism since µ = ker φ is biorder-preserving. 4.5. IDEMPOTENTS IN TE 90

The following is a useful corollary. Its proof is due to Jordan and appears in [23] Corollary 4.2.3.

Corollary 4.18. Let E be any boset and α : ω(e) → ω(eα), β : ω(f) → ω(fβ) principal

ideal isomorphisms of E. Then e f and eα fβ if and only if φα H φβ.

Proof. By Theorem 4.17 and remarks preceding Proposition 2.1 we have

◦ e f ⇐⇒ φe φf ⇐⇒ φe R φf and ◦ eα fβ ⇐⇒ φeα φfβ ⇐⇒ φeα L φfβ. ◦ ◦ Note also that φe R φα L φeα, since by Proposition 4.10 we have

φeφα = φα, φαφα−1 = φe, φαφeα = φα, and φα−1 φα = φeα. Suppose first e f and eα fβ, then by the above we have

◦ ◦ ◦ ◦ ◦ ◦ φα R φe R φf R φβ and φα L φeα L φfβ L φβ.

Conversely, suppose now φα H φβ. Then

◦ ◦ ◦ ◦ ◦ ◦ φe R φα R φβ R φf and φeα L φα L φβ L φfβ, which implies that e f and eα fβ. 

The following lemma is due to Hall and its proof appears in [5] Lemma 4.

Lemma 4.19. Let E be any boset, e, f ∈ E, and α any idempotent in SE such that

φe R α L φf . Then α = φx for some x ∈ E.

Easdown remarks in [5] that Hall proved Lemma 4.19 in order to show that any idempotent

in SE that is D-related to an element of Eφ must be equal to an element of Eφ. We provide our own proof in the next two lemmas where D is interpreted to be the equivalence hR ∪ L i = h ∪ i on the boset E. 4.5. IDEMPOTENTS IN TE 91

Lemma 4.20. Let E be any boset, e ∈ E, and α any idempotent in SE. Then

(i) α R φe ⇒ α = φx for some x ∈ E; and dually

(ii) α L φe ⇒ α = φx for some x ∈ E.

Proof. By duality it is sufficient to prove (i). There exists x1,...,xn ∈ E such that α =

φx1 ...φxn . We have Leρe = Le =6 ∞ which implies Leρx1 ...ρxn =6 ∞ since ρe R ρx1 ...ρxn .

This implies that there exists y1,...,yn ∈ E such that

x1 xn

e y1 y1x1 ··· yn ynxn.

Since φ is a homomorphism and by Green’s Lemma we have the following egg-box diagram.

φe φx1 ··· φxn . . . .

φy1 ··· φy1 φx1 ··· φy1 φx1 φx2 φy1 φx1 ··· φxn−1 ··· φy1 φx1 ··· φxn . . . .

φy2 ··· φy2 φx2 . ..

φyn−1 φxn−1 . . . .

φyn ··· φyn φxn = φynxn

The result follows by Lemma 4.19. 

Lemma 4.21. Let E be any boset, e ∈ E, and α any idempotent in SE such that α D φe.

Then α = φx for some x ∈ E.

Proof. There exists idempotents α1,...,αn ∈ SE such that

φe A1 α1 A2 ··· An αn = α

where A1,..., An ∈{R, L }. The proof is by induction on n. Suppose first n = 1. Then the result holds by Lemma 4.21. Suppose now m> 1 and, as an inductive hypothesis, the ′ result holds for n = m − 1. Then αm−1 = φx′ for some x ∈ E by the inductive hypothesis,

and φx′ Am αm. By Lemma 4.21 αm = φx for some x ∈ E, so the result holds for n = m. The result follows by induction.  4.6. TE FOR REGULAR BOSETS 92

4.6. TE for Regular Bosets

In this section we investigate some properties of the semigroups TE and SE where E is a regular boset.

The following theorem appears in Jordan [23] as Corollary 3.4.7 and Theorem 3.4.8. It shows that, as in the case of the Munn semigroup, the angular brackets can be replaced by curly brackets in the definition of TE whenever E is a regular boset.

Theorem 4.22. Let E be any regular boset. Then

TE = {φα | α is a principal ideal isomorphism of E} is a regular semigroup. 

The proof is quite involved and technical and shows that for any principal ideal isomor- phisms α : ω(e) → ω(eα) and β : ω(f) → ω(fβ) on an arbitrary boset E, there is a principal ideal isomorphism γ : ω(g) → ω(gγ) on E such that φαφβ = φγ if and only if the sandwich set S(eα, f) is nonempty. This proof appears in [23] Theorem 3.4.6. The difficulty of Jordan’s proof highlights the vastness of Nambooripad’s groundbreaking work [28] which the proof is reconstructing.

The remaining results in this section describe the bosets of idempotents of the semigroups

SE and TE.

The first result is due to Easdown [7] Theorem 3.

∼ Theorem 4.23. Let E be any regular boset. Then E(SE) = E. 

The proof involves showing that each idempotent α ∈ SE is equal to φe for some e ∈ E, for then the result follows by Theorem 4.17. This is achieved by showing that there exists x, y ∈ E such that φx R α L φy. The result then follows by Lemma 4.19.

This result was extended by Jordan [23] Theorem 3.5.3 to TE to complete the reconstruc- tion of Nambooripad’s result.

∼ Theorem 4.24. Let E be any regular boset. Then E(TE) = E.  4.6. TE FOR REGULAR BOSETS 93

The proof involves showing that for any principal ideal isomorphism α : ω(e) → ω(eα)

on an arbitrary boset E such that φα is idempotent, there is an element f ∈ E such that e f eα and φα = φf . This proof can be found in [23] Lemma 3.5.2.

This section is finished by illustrating the last two results with some examples. This will

be done by describing the bosets of idempotents of SE and TE for the boset of idempotents of a matrix semigroup, and also for the sawtooth bosets in Section 3.5. First we consider some regular examples.

Let F be any field and consider the semigroup

S = Matn(F)= {n × n matrices over F}

under matrix multiplication. Then

µ = {(A, B) ∈ S × S | (∃k ∈ F \{0}) B = kA}

is the maximum idempotent separating congruence on S. This follows quickly from the fact that S is regular, so the maximum idempotent separating congruence is contained in

H (see Corollary 4.4). The quotient S/µ is then fundamental. When F = Z2 we have

µ = 1, so S = Matn(Z2) is fundamental. Consider the example

S = Mat2(Z2).

The egg-box diagram of S appears in Figure 4.4. The boset diagram of E(S) appears in

Figure 4.5. The automorphism group of the whole diagram is S3, and the automorphism

groups of the smaller principal ideals are trivial. The egg-box diagram for TE where E is this boset appears in Figure 4.6. The rows and columns of the middle D-class are indexed by the nontrivial R-classes and L -classes of E respectively. By Nambooripad’s theory we

must have S = TE (see Theorem 5.4). Note that TE has boset of idempotents E. Since

E ⊆ E(SE) ⊆ E(TE)= E, the boset of idempotents of SE must also be E.

Let E1 be the regular boset given by the diagram in Figure 4.7. The semigroup SE1 is a

five element band in which φeφf = φf φe = φg. The semigroup TE1 contains nonidentity

isomorphisms between the principal ideals ω(e) and ω(f). The egg-box diagrams for SE1

and TE1 appear in Figure 4.8 and Figure 4.9 respectively.

Now we demonstrate that these results do not necessarily hold for nonregular bosets. Let

E2 be the nonregular boset defined by the diagram in Figure 4.10. The semigroup SE2

is a seven element band with two new idempotents φeφf φe = φf φe and φf φeφf = φeφf .

The semigroup TE2 contains products of isomorphisms between the principal ideals ω(e) and ω(f). The egg-box diagrams for SE2 and TE2 appear in Figure 4.11 and Figure 4.12 4.6. TE FOR REGULAR BOSETS 94

1 0 ∼ = S3 " 0 1 #

1 0 1 1 " 0 0 # " 0 0 #

0 0 0 0 " 1 1 # " 0 1 #

1 0 0 1 ∼ = S1 " 1 0 # " 0 1 #

0 0 ∼ = S1 " 0 0 #

Figure 4.4. Egg-box diagram of Mat2(Z2)

respectively. Note that the bosets of idempotents of SE2 and TE2 both contain seven elements, which is strictly larger than E2.

Let E3 be the boset given by the diagram in Figure 4.13. The semigroup SE3 is a six element semigroup which contains five idempotents and one non-regular element, which is the product φf φe. In TE3 the product φf φe becomes regular and lies in a D-class with three new idempotents. The egg-box diagrams for SE3 and TE3 appear in Figure 4.14 and Figure 4.15 respectively. Note that we use the convention that an arrow from an idempotent x in the lowest D-class to an L -class L in the D-class above means that x is a zero for every idempotent in L. This convention may be used later without comment. Observe that the boset of idempotents of SE3 is isomorphic to E3, but the boset of idempotents of

TE3 is strictly larger than E3. 4.7. TE FOR POSETS 95

Figure 4.5. Boset diagram of E(Mat2(Z2))

4.7. TE for Posets

Let E be any poset under ≤. Then E becomes a boset by defining = = ≤ and ef = fe = e whenever e ≤ f. Since = , we can identify e ∈ E with Re = Le = {e}. Observe that, if α : ω(e) → ω(eα) is any principal ideal isomorphism on E then

ρα : x 7→ xα if x e and −1 λα : y 7→ yα if y eα. −1 This shows that ρα and λα are the principal ideal isomorphisms α and α , respectively, in disguise. We can therefore make the identification

∗ φα =(ρα,λα) ≡ α and

TE ≡hα | α is a principal ideal isomorphism of Ei.

In practice, we often replace ‘≡’ with ‘=’ and think of elements of TE as products of principal ideal isomorphisms, so

TE = hα | α is a principal ideal isomorphism of Ei . IE.

Let α1 ··· αn ∈ TE where α1,...,αn are principal ideal isomorphisms on E. Then the −1 −1 inverse αn ··· α1 of α1 ··· αn is in TE, so TE is regular. Also, idempotents commute in 4.7. TE FOR POSETS 96

∼ = S3

∼ = S1

∼ = S1

Figure 4.6. Egg-box diagram of TE(Mat2(Z2))

e f

h g g k hf k ke

Figure 4.7. Boset diagram of E1 4.7. TE FOR POSETS 97

∼ φe φf ∼ S1 = = S1

∼ = S1 φh φk φg

Figure 4.8. Egg-box diagram of SE1

φe

φf ∼ = S2

∼ = S1 φh φk φg

Figure 4.9. Egg-box diagram of TE1

e f

h g k k k ke hf

Figure 4.10. Boset diagram of E2 4.7. TE FOR POSETS 98

∼ ∼ S1 = φe φf = S1

∼ φf φe φeφf = S1

∼ φh φk = S1 φg

Figure 4.11. Egg-box diagram of SE2

φe

∼ φf = S2

∼ φf φe φeφf = S2

∼ φh φk = S1 φg

Figure 4.12. Egg-box diagram of TE2 4.7. TE FOR POSETS 99 e f

h g k k k ke hf

Figure 4.13. Boset diagram of E3

∼ ∼ S1 = φe φf = S1

Non-regular

φf φe ← singleton D-class

∼ φh φk = S1 φg

Figure 4.14. Egg-box diagram of SE3

TE since they do in IE. This shows that TE is an inverse semigroup with semilattice of

idempotents F = E(TE). In this case

SE = h1ω(e) | e ∈ Ei.

For each e, f ∈ E we have

1ω(e)1ω(f) =1ω(e)∩ω(f) =1ω(f)1ω(e), in particular, if e = f then

1ω(e)1ω(e) =1ω(e)∩ω(e) =1ω(e). 4.7. TE FOR POSETS 100

φe

∼ φf = S2

φf φe

∼ = S2

∼ φh φk = S1 φg

Figure 4.15. Egg-box diagram of TE3

This shows that SE is a commutative semigroup of idempotents, which is a semilattice. ∼ ∼ Then E(SE) = SE so SE(SE ) = SE. The relationship between TE and TE(TE ) is much more delicate, but we have the following result.

Theorem 4.25. Let E be any poset and F = E(TE). Then TE embeds in TF .

Proof. Let α : ω(e) → ω(eα) be a principal ideal isomorphism on E and define a function

−1 α : ω(1ω(e)) → ω(1ω(eα)), γ 7→ α γα.

−1 −1 Since αα = 1ω(e) and α α = 1ω(eα), α is the principal ideal isomorphism θα,α−1 from Section 4.3 in disguise. Define a function

φ : TE → TF , α1 ··· αn 7→ α1 ··· αn.

Then φ is clearly a semigroup homomorphism, provided that it is well defined. Suppose

α1 ··· αn = β1 ··· βm for some principal ideal isomorphisms α1,...,αn, β1,...,βm of E. 4.7. TE FOR POSETS 101

Then −1 −1 −1 −1 −1 −1 αn ··· α1 =(α1 ··· αn) =(β1 ··· βm) = βm ··· β1 .

If γ ∈ dom(α1 ··· αn) then

−1 −1 −1 −1 γα1 ··· αn = αn ··· α1 γα1 ··· αn = βm ··· β1 γβ1 ··· βm = γβ1 ··· βm. By symmetry this is enough to prove

α1 ··· αn = β1 ··· βm, that is,

(α1 ··· αn)φ =(β1 ··· βm)φ, so φ is a well defined homomorphism. It remains to prove that φ is one-one. Suppose

(α1 ··· αn)φ =(β1 ··· βm)φ for some principal ideal isomorphisms α1,...,αn, β1,...,βm of E. Then

α1 ··· αn = β1 ··· βm.

Let x ∈ dom(α1 ··· αn), so y = xα1 ··· αn is defined. But

1ω(x)α1 ··· αn =1ω(x)β1 ··· βm, so −1 −1 −1 −1 αn ··· α1 1ω(x)α1 ··· αn = βm ··· β1 1ω(x)β1 ··· βm. In particular

−1 −1 −1 −1 yβm ··· β1 1ω(x)β1 ··· βm = yαn ··· α1 1ω(x)α1 ··· αn = y, so −1 −1 −1 −1 yβm ··· β1 1ω(x) = yβm ··· β1 . −1 −1 Put z = yβm ··· β1 , so z1ω(x) = z, that is, z ∈ ω(x). Also

1ω(z)α1 ··· αn =1ω(z)β1 ··· βm, so −1 −1 −1 −1 αn ··· α1 1ω(z)α1 ··· αn = βm ··· β1 1ω(z)β1 ··· βm, giving −1 −1 −1 −1 yαn ··· α1 1ω(z)α1 ··· αn = yβm ··· β1 1ω(z)β1 ··· βm = y, so −1 −1 −1 −1 yαn ··· α1 1ω(z) = yαn ··· α1 , that is x1ω(z) = x, so x ∈ ω(z). Hence x = z and

xα1 ··· αn = y = zβ1 ··· βm = xβ1 ··· βm. By symmetry this is sufficient to prove that

α1 ··· αn = β1 ··· βm, 4.7. TE FOR POSETS 102 a b

c d e

f g f ′ g′

Figure 4.16. Hasse diagram of the poset E

1ω(a) 1ω(b)

1ω(c) 1ω(d) α 1ω(e)

1ω(f) 1ω(g) 1ω(f ′) 1ω(g′)

ζ

Figure 4.17. Hasse diagram of the semilattice F = E(TE) for E of Figure 4.16 so φ is one-one. 

We illustrate this result with an example of a poset for which the embedding is strict. Let E be the poset defined by the Hasse diagram in Figure 4.16.

Then ω(a) ∼= ω(b), ω(d) =∼ ω(e), ω(f) ∼= ω(g) ∼= ω(f ′) ∼= ω(g′) and

F = {1ω(x) | x ∈ E}∪{α,ζ}, 4.7. TE FOR POSETS 103

1ω(a)

1ω(b) ∼ = S1

1ω(d)

1ω(c) ∼ ∼ S2 = α = S2 1ω(e) ∼ = S1

1ω(f)

1ω(g)

1ω(f ′)

∼ = S1 1ω(g′)

∼ = S1 ζ

Figure 4.18. Egg-box diagram of TE for E of Figure 4.16 4.7. TE FOR POSETS 104

1ω(a)

1ω(b) ∼ = S1

1ω(d) α

1ω(e) 1ω(c) ∼ ∼ = S1 = S2

1ω(f)

1ω(g)

1ω(f ′)

∼ = S1 1ω(g′)

∼ = S1 ζ

Figure 4.19. Egg-box diagram of TF for F of Figure 4.17 4.8. TE FOR SEMILATTICES 105

where α =1ω(a)1ω(b) =1{f,g} and ζ =1ω(f)1ω(g) =1∅. The Hasse diagram for the semilattice F appears in Figure 4.17. The isomorphic principal ideals of F are ∼ ∼ ω(1ω(a)) = ω(1ω(b)), ω(1ω(d)) = ω(1ω(e)),

∼ ∼ ′ ∼ ′ ∼ ω(1ω(f)) = ω(1ω(g)) = ω(1ω(f )) = ω(1ω(g )) and ω(α) = ω(1ω(c)).

The new elements in TF are isomorphisms between the principal ideals ω(α) and ω(1ω(c)).

The egg-box diagrams for TE and TF appear in Figure 4.18 and Figure 4.19 respectively.

It remains an open problem to determine for which posets the embedding is strict.

4.8. TE for Semilattices

Let E be any semilattice. Then E is a poset which is regular, so all of the results from Section 4.6 and Section 4.7 apply.

By Theorem 4.22, Theorem 4.24 and the identification in Section 4.7,

TE = {α | α is a principal ideal isomorphism of E} is an inverse semigroup with semilattice of idempotents isomorphic to E. This is precisely the definition of the Munn semigroup of a semilattice E.

Also, if S is any inverse semigroup then the representation φ, with the identification made in Section 4.7, becomes

φ : S → TE, a 7→ θa,a−1 where −1 −1 −1 θa,a−1 : ω(aa ) → ω(a a), x 7→ a xa which is precisely the definition of Munn’s representation of S by TE. CHAPTER 5

TE and Regular-Generated Semigroups

In this chapter we prove a main theorem of this thesis that every fundamental regular- generated semigroup with boset of idempotents E embeds in TE, and conversely every regular-generated subsemigroup of TE with boset of idempotents Eφ is fundamental. This result demonstrates the importance of TE, since every fundamental regular-generated semi- group can be reconstructed as a subsemigroup of TE where E is its boset of idempotents.

We also prove another main result of this thesis, that TE is a fundamental semigroup for any boset E. These results generalise Munn’s result that TE is the maximum fundamental inverse semigroup with semilattice of idempotents E. The consequences of these results is also investigated for TE which is itself a fundamental regular-generated semigroup.

5.1. Arbitrary Regular-Generated Semigroups

Let S be any semigroup with boset of idempotents E, and R any subset of Reg(S) which contains E. Then E ⊆ E(hRi) ⊆ E(S)= E, so E(hRi)= E. Consider the semigroup homomorphism

∗ φ : hRi → TE/L ∪{∞} × TE/R∪{∞}, a 7→ φa.

Suppose that a ∈hRi. Then a = r1 ··· rn for some r1,...,rn ∈ R, so that, by Proposition 4.9,

aφ = φa = φr1···rn = φr1 ··· φrn ∈ TE,

which shows that im φ ⊆ TE. By the fundamental homomorphism theorem we have ∼ hRi/µ(hRi) = im φ ⊆ TE.

In particular, if R = E then a = e1 ··· en for some e1,...,en ∈ E and

aφ = φa = φe1···en = φe1 ··· φen ∈ SE.

Conversely, suppose that α ∈ SE. Then

α = φe1 ··· φen = φe1···en =(e1 ··· en)φ 106 5.2. THE CONTRAST BETWEEN THE SEMIGROUPS hEi AND SE = hEφi 107 e f

g h

Figure 5.1. Rectangular boset diagram

for some e1,...,en ∈ E. This shows that im φ = SE. By the fundamental homomorphism theorem we have ∼ hEi/µ(hEi) = im φ = SE. We have proven the following.

Theorem 5.1. Let S be any regular-generated semigroup with boset of idempotents E.

Then im φ ⊆ TE and S/µ embeds in TE. In particular, if S is idempotent-generated, then ∼ im φ = SE and SE = S/µ is fundamental. Thus, for any boset E there is at most one fundamental idempotent-generated semigroup with boset of idempotents E. 

The last claim of Theorem 5.1, that is there is at most one fundamental idempotent- generated semigroup with boset of idempotents E, was proven by Hall [20] Corollary 17 in the case where E is a regular boset. Hall’s proof uses a construction based on the partial groupoid of idempotents of a regular semigroup which is generated by its idempotents.

5.2. The Contrast between the Semigroups hEi and SE = hEφi

∼ Observe that although E = Eφ as bosets, hEi and SE = hEφi can be very different semigroups. We illustrate this with some examples.

Let E be the boset defined by the diagram in Figure 5.1. Note that this completely defines the boset since all products are trivial. Consider the free semigroup FE on the boset E.

Elements of FE can be represented uniquely by words of alternating e’s and h’s, and words of alternating f’s and g’s. By Proposition 2.4 we must have

fg ∈ He, eh ∈ Hf , he ∈ Hg and gf ∈ Hh.

This implies, again by Proposition 2.4 that

ehe =(eh)(he) ∈ He, fgf ∈ Hf , gfg ∈ Hg and heh ∈ Hh. 5.2. THE CONTRAST BETWEEN THE SEMIGROUPS hEi AND SE = hEφi 108

∼ e f ∼ C∞ = hfgi = = hehi = C∞

∼ ∼ C∞ = hhei = = hgfi = C∞ g h

Figure 5.2. Egg-box diagram of FE for E of Figure 5.1

∼ φf φg = φe φf = φeφh ∼ C1 = = C1

∼ ∼ C1 = = C1 φhφe = φg φh = φgφf

Figure 5.3. Egg-box diagram of SE for E of Figure 5.1

n m Consider the group He. Elements of He are the words (fg) for n ≥ 1 and e(he) for m ≥ 0. Also, fg and ehe are mutual group inverses since

fgehe = fghe = fhe = fe = e and ehefg = ehfg = ehg = eg = e, so He is an infinite cyclic group generated by fg. Similarly Hf , Hg and Hh are infinite cyclic groups generated by eh, he and gf respectively. The egg-box diagram for FE appears in Figure 5.2. In contrast, the semigroup SE contains only four elements. For each x ∈ E we have

ρx : L 7→ Lx and

λx : R 7→ Rx. It follows quickly that

φf φg = φe, φeφh = φf , φhφe = φg and φgφf = φh.

The egg-box diagram for SE appears in Figure 5.3. In fact both FE and SE are isomorphic 5.2. THE CONTRAST BETWEEN THE SEMIGROUPS hEi AND SE = hEφi 109

∼ ∼ S1 = e f = S1

Infinitely many fe ef non-regular ← singleton efe fef H -classes

. .

∼ h k = C1 g

Figure 5.4. Egg-box diagram of FE2

to Rees matrix semigroups M(G;2, 2; P ) where

1 1 P = " 1 1 # and G is an infinite cyclic and trivial group respectively.

Consider the free semigroup FE2 on the nonregular boset E2 given by the diagram in Figure

4.10. Elements of FE2 can be uniquely represented by the singleton words g, h and k, and words of alternating e’s and f’s. The semigroup FE2 consists of singleton H -classes. The

five H -classes He, Hf , Hg, Hh and Hk are trivial groups. The other H -classes each contain a single word of alternating e’s and f’s which is nonregular. The egg-box diagram for FE2 appears in Figure 5.4. Recall that the semigroup SE2 is a seven element band with two new idempotents φf φe = φeφf φe and φeφf = φf φeφf . Its egg-box diagram appears in Figure 4.11. 5.3. RECONSTRUCTING FUNDAMENTAL REGULAR GENERATED SEMIGROUPS 110 1

··· ← elements of G

Figure 5.5. Boset diagram of E = E(G ⊔ G)

∼ 1 = SG

∼ ··· = S1

Figure 5.6. Egg-box diagram of TE(G⊔G) ≡ SG ⊔ G

5.3. Reconstructing Fundamental Regular Generated Semigroups

Theorem 5.1 also shows that every fundamental regular generated semigroup can be re- constructed from its boset of idempotents. Consider the following example. Let G be any group and G ⊔ G the fundamental semigroup in Figure 2.6. The semigroup G ⊔ G has boset of idempotents E with diagram in Figure 5.5. The principal ideals of E are the whole boset ω(1) = E and the singleton ideals ω(g)= {g} for each g ∈ G. It follows quickly from the definition of φ that if g, h ∈ G and α is the unique principal ideal isomorphism from

ω(g) to ω(h) then φα = φh. The automorphism group of ω(1) is isomorphic to SG since automorphisms of E correspond to permutations of G. By Proposition 4.10 we have the egg-box diagram for TE in Figure 5.6. Let g ∈ G. By Cayley’s Theorem G embeds in SG, so we can identify g with its image, which is the mapping

G → G, x 7→ xg. 5.4. REGULAR-GENERATED SUBSEMIGROUPS OF TE WITH BOSET OF IDEMPOTENTS Eφ 111

The corresponding element of SG,

G → G, x 7→ xg

can be extended to a principal ideal isomorphism

αg : ω(1) → ω(1), 1 7→ 1, x 7→ xg.

It follows quickly from Proposition 4.10 and the definition of φαg that, for each g, h ∈ G

φαg φαh = φαgαh = φαgh ,

φαg φh = φgφh = φh,

φgφαh = φgαh = φgh. This shows that the mapping

G ⊔ G → TE, g 7→ φαg , g 7→ φg

is an injective homomorphism, so G⊔G embeds in TE. Hence, in the sense of being embed- ded inside the “synthetic” TE, G ⊔ G can be reconstructed from its boset of idempotents.

5.4. Regular-Generated Subsemigroups of TE with Boset of Idempotents Eφ

Theorem 5.1 shows that every fundamental regular-generated semigroup with boset of idempotents E is isomorphic to a subsemigroup of TE with boset of idempotents Eφ. It is natural to ask the question “is every regular-generated subsemigroup of TE with boset of idempotents Eφ fundamental?” The answer turns out to be yes, but to prove this we first need a lemma, the proof of which is motivated by the technique of Jordan [23] Proposition 3.6.8.

Lemma 5.2. Let E be any boset and S any subsemigroup of TE with boset of idempo- tents Eφ, so that idempotents in S can be written in the form φe and we can make the identification φe ≡ e. Then φa ≡ a for every regular element a ∈ S.

Proof. Let a ∈ Reg(S) and a′′ ∈ V (a). Then there are some principal ideal isomorphisms

αi : ω(ei) → ω(eiαi) for i = 1,...,n and βj : ω(fj) → ω(fjβj) for j = 1,...,m such ′′ ′′ that a = φα1 ··· φαn and a = φβ1 ··· φβm . There exists e, g ∈ E such that aa = φe and ′′ a a = φg. This implies that Leρα1 ··· ραn ρβ1 ··· ρβm = Leρe = Le which in turn implies that Leρα1 ··· ραn =6 ∞ so there exists x1 ∈ Le, x2,...,xn ∈ E such that 5.4. REGULAR-GENERATED SUBSEMIGROUPS OF TE WITH BOSET OF IDEMPOTENTS Eφ 112

e1 e1α1 e2 e2α2

e x1 x1e1 (x1e1)α1 x2 x2e2 (x2e2)α2 ···

en enαn

··· xn xnen (xnen)αn.

By Lemma 4.16 there is a principal ideal isomorphism α : ω(e) → ω(f) where f =(xnen)αn ′′ ′′ such that φα = φeφα1 ··· φαn = aa a = a. In S we have aa = φe, φea = a, aφf = a and ′′ ′′ ′′ φf a a = φα−1 φαa a = φα−1 aa a = φα−1 a = φα−1 φα = φf so φe R a L φf so there exists ′ ′ a ∈ V (a) such that φe L a R φf . In TE we have the egg-box diagram

a = φα ··· φe . . . . ′ φf ··· a ,φα−1

′ ′ and then a = φα−1 by uniqueness of inverses in H -classes. Note also that aa = φe, ′ a a = φf and, for any h ∈ E, φh φe implies h e. By Lemma 4.13 we have

θa,a′ : ω(φe) → ω(φf ), φh 7→ φα−1 φhφα = φα−1 φαφhα = φhα.

Under the identification of idempotents,

θa,a′ : ω(e) → ω(f), h 7→ hα,

′ so θa,a ≡ α which implies that φa = φθa,a′ ≡ φα = a by Proposition 4.9. 

Let E be any boset and S any regular-generated subsemigroup of TE with boset of idem-

potents Eφ. Let a, b ∈ S such that a µ b. Then a = a1 ··· an and b = b1 ··· bm for some

a1,...,an, b1,...,bm ∈ Reg(S). By Lemma 5.2 and since µ = ker φ we have

a = a1 ··· an ≡ φa1 ··· φan = φa1···an = φa = φb = φb1···bm = φb1 ··· φbm ≡ b1 ··· bm = b.

This shows that µ is the trivial congruence on S, so S is fundamental. Combining these observations with Theorem 5.1 we have now proven the following theorem.

Theorem 5.3. Let S be any regular-generated semigroup with boset of idempotents E.

Then S is fundamental if and only if it is isomorphic to a subsemigroup of TE with boset of idempotents Eφ. In particular, if S is idempotent-generated then it is fundamental if

and only if it is isomorphic to SE.  5.5. TE IS FUNDAMENTAL 113

If S is a regular semigroup then by Theorem 3.8, Theorem 4.23 and Theorem 4.24 this result becomes the following.

Theorem 5.4. Let S be any regular semigroup with boset of idempotents E. Then S is fundamental if and only if it is isomorphic to a full subsemigroup of TE which contains SE as a full subsemigroup. 

This result was proved by Jordan [23] Theorem 3.6.9 and shows that TE is the maximum fundamental regular semigroup with boset of idempotents E, whenever E is a regular boset. This reproduces results of Hall [20], Grillet [18, 19], Nambooripad [26, 28] and Clifford [1], who also construct maximum fundamental regular semigroups.

5.5. TE is Fundamental

In this section we prove that TE is fundamental for any boset E. Note that if E is a regular boset then TE has boset of idempotents Eφ by Theorem 4.24, so TE is fundamental by Theorem 5.3. If E is a nonregular boset then we have seen in Section 4.6 that the boset of idempotents of TE can be larger than Eφ, so TE is not guaranteed to be a fundamental semigroup by Theorem 5.3.

First we investigate the property that SE is fundamental for any boset E. By Theorem 5.1,

SE is fundamental for any boset E. The proof however uses the fundamental representation

φ. We provide an alternative proof which uses the definition of TE directly.

Theorem 5.5. Let E be any boset. Then SE is a fundamental semigroup.

Proof. It is sufficient to prove that the congruence µ on the semigroup SE is the trivial congruence, for then SE is fundamental by Theorem 4.7 and we will be done.

Let a, b ∈ SE such that a µ b. It is enough to show that a = b, for then µ is the trivial congruence. There exists e1,...,en, f1,...,fm ∈ E such that a = φe1 ··· φen and b =

φf1 ··· φfm . By duality it is sufficient to prove ρe1 ··· ρen = ρf1 ··· ρfm . Let L be any

L -class. Suppose Lρe1 ··· ρen =6 ∞. Then there exists x1 ∈ L, x2,...,xn ∈ E such that

e1 en

x1 x1e1 ··· xn xnen 5.5. TE IS FUNDAMENTAL 114

and Lρe1 ··· ρen = Lxnen . By Theorem 4.17 we have

φe1 φen

φx1 φx1e1 ··· φxn φxnen

and L ρ ··· ρ = L . Since µ = kerφ we have φx1 φe1 φen φxnen φ ··· φ = φ = φ = φ = φ = φ ··· φ , φe1 φen φe1 ···φen a b φf1 ···φfm φf1 φfm so L ρ ··· ρ = L ρ ··· ρ = L φx1 φf1 φfm φx1 φe1 φen φxnen and there exists α ∈ L , α ,...,α ∈ E(S ) such that 1 φx1 2 m E

φf1 φfm

φx1 α1 α1φf1 ··· αm αmφfm φxnen .

By Lemma 4.21 there exists y1,...,ym ∈ E such that

αi = φyi and

αiφfi = φyi φfi = φyifi for each i =1,...,m. Again by Theorem 4.17 we have

f1 fm

x1 y1 y1f1 ··· ym ymfm xnen

and

Lρf1 ··· ρfm = Lymfm = Lxnen = Lρe1 ··· ρen .

By symmetry this is enough to prove ρe1 ··· ρen = ρf1 ··· ρfm . 

We now prove the main theorem.

Theorem 5.6. Let E be any boset. Then TE is a fundamental semigroup.

Proof. It is sufficient to prove that the congruence µ on the semigroup TE is the trivial congruence, for then TE is fundamental by Theorem 4.7 and we will be done. 5.5. TE IS FUNDAMENTAL 115

Let a, b ∈ TE such that a µ b. It is enough to show that a = b, for then µ is the trivial congruence. There exists principal ideal isomorphisms of E

αi : ω(ei) → ω(eiαi) for i =1,...,n

and

βj : ω(fj) → ω(fjβj) for j =1,...,m

such that a = φα1 ··· φαn and b = φβ1 ··· φβm . By duality it is sufficient to prove ρα1 ··· ραn =

ρβ1 ··· ρβm . Let L be any L -class. Suppose Lρα1 ··· ραn =6 ∞. Then there exists x1 ∈

L, x2,...,xn ∈ E such that

e1 e1α1 e2 e2α2

x1 x1e1 (x1e1)α1 x2 x2e2 (x2e2)α2 ···

en enαn

··· xn xnen (xnen)αn.

By Lemma 4.16 there is a principal ideal isomorphism

γ : ω(x1) → ω((xnen)αn) of E such that

φx1 φα1 ··· φαn = φγ, so

−1 −1 φx1 φα1 ··· φαn φγ = φγφγ = φx1 .

◦ This shows that φx1 R φx1 φα1 ··· φαn , so by definition of µ we have

◦ (5.7) φx1 R φx1 φβ1 ··· φβm

and

(5.8) φx1 φα1 ··· φαn H φx1 φβ1 ··· φβm .

By (5.7) we have

Lρβ1 ··· ρβm = Lρx1 ρβ1 ··· ρβm =6 ∞

since Lρx1 = L =6 ∞, so there exists y1 ∈ L, y2,...,ym ∈ E such that 5.6. EMBEDDINGS OF TE 116

f1 f1β1 f2 f2β2

x1 y1 y1f1 (y1f1)β1 y2 y2f2 (y2f2)β2 ···

fm fmβm

··· ym ymfm (ymfm)βm.

By Lemma 4.16 there is a principal ideal isomorphism

δ : ω(x1) → ω((ymfm)βm)

of E such that

φx1 φβ1 ··· φβm = φδ, so

φγ H φδ by (5.8) which implies

(xnen)αn (ymfm)βm by Corollary 4.18. This shows that

Lρα1 ··· ραn = L(xnen)αn = L(ymfm)βm = Lρβ1 ··· ρβm .

By symmetry this is enough to prove ρα1 ··· ραn = ρβ1 ··· ρβm . 

5.6. Embeddings of TE

Let E be any boset and put F = E(TE) and G = E(SE). The semigroup TE is regular generated, so by Theorem 5.1 and Theorem 5.6 we have ∼ TE = TE/µ . TF .

The semigroup SE is idempotent generated, so again by Theorem 5.1 we have ∼ ∼ SE = SE/µ = SG.

We have proven the following.

Corollary 5.9. Let E be any boset and put F = E(TE) and G = E(SE). Then TE embeds ∼ in TF and SE = SG.  5.6. EMBEDDINGS OF TE 117 ∼ ∼ Note that if E is regular then, by Nambooripad’s result E = F , so TE = TF . Any example in which the embedding is strict must involve a nonregular boset.

Let E be any boset and put E1 = E and En = E(TEn−1 ) for n> 1. Then TEn . TEn+1 for all n ≥ 1 so we have

TE1 . TE2 . ··· . TEn . TEn+1 . ··· + ∼ ∼ Note that if En is a regular boset for any n ∈ Z then En = Em and hence TEn = TEm for all m ≥ n. It remains an open problem to either find a nonregular boset E for which + En is a nonregular boset and the embedding TEn . TEn+1 is strict for all n ∈ Z , or to + prove that for any boset E there exists n ∈ Z such that the embeddings TEm . TEm+1 are isomorphisms for all m ≥ n. CHAPTER 6

TE for Sawtooth Bosets

If E is a nonregular boset then SE and TE may be regular or nonregular. In this chapter we investigate the regularity of SE and TE for sawtooth bosets. A class of sawtooth bosets, called cyclic sawtooth bosets, which contain infinitely many examples of regular

and nonregular bosets is introduced for which SE and TE are always regular. Regularity of SE and TE for arbitrary sawtooth bosets with 2 teeth is also explored. The regularity

criteria are due to Jordan. The criterion for SE appears in [23] and is proved here using the theory of cyclic sawtooth bosets. The criterion for TE appears erroneously in [23]. The correct criterion appears in this chapter.

6.1. Preliminaries

Let E be any boset in which = . Then L -classes are singletons, so each L - class can be identified with its unique element. The following result is due to Jordan [23]

Proposition 4.3.1 and is a nice simplification of TE.

Proposition 6.1. Let E be any boset in which = . Then ∼ TE = hρα | α is a principal ideal isomorphism of Ei.



The proof involves showing that λαn ··· λα1 = λβm ··· λβ1 whenever ρα1 ··· ραn = ρβ1 ··· ρβm for any principal ideal isomorphisms α1,...,αn, β1,...,βm of E, for then the onto homo- morphism

π : TE →hρα | α is a principal ideal isomorphism of Ei, φα1 ··· φαn 7→ ρα1 ··· ραn where α1,...,αn are any principal ideal isomorphisms of E, is also one-one.

We can therefore make the identification

φα1 ··· φαn ≡ ρα1 ··· ραn 118 6.1. PRELIMINARIES 119 and

TE ≡hρα | α is a principal ideal isomorphism of Ei.

In practice, it is usual to replace “≡” with “=” and think of elements of TE as products of ρ’s so that

TE = hρα | α is a principal ideal isomorphism of Ei.

Let E be any sawtooth boset. Recall from Section 3.5 that E is the disjoint union of two subsets T and R. The rest of this section includes useful observations about elements of

TE that will often be used without comment later in the chapter.

Let r ∈ R. It follows quickly from the fact that sr = r for all s ∈ R and the definition of

TE that ρr is given by

ρr : t 7→ ∞, s 7→ r

for all t ∈ T and s ∈ R, so ρr may be considered as a constant mapping with domain R.

If α : ω(r) → ω(rα) is a principal ideal isomorphism with r, rα ∈ R then ρα = ρrα. The

elements ρr for r ∈ R form an R-class, so R is reproduced in SE and TE.

Let α : ω(t) → ω(tα) be a principal ideal isomorphism with t, tα ∈ T . Then ρα is given by

ρα : t 7→ tα, u 7→ ∞, r 7→ (rt)α for all u ∈ T \{t} and r ∈ R. These elements lie in square D-classes whose rows and columns are indexed by {t ∈ T ||ω(t)| = |X|}

for some set X. The automorphism groups are isomorphic to SX . b

The rest of TE is described by the next result which is due to Jordan and appears in [23] Lemma 4.3.3 and Lemma 4.3.4.

Lemma 6.2. Let E be any sawtooth boset and

θ ∈ TE \{ρα | α is a principal ideal isomorphism of E}.

Then there is an integer n ≥ 2, and principal ideal isomorphisms αi : ω(ti) → ω(tiαi) with

ti, tiαi ∈ T for i =1,...,n such that

θ = ρα1 ··· ραn and

tiαi =6 ti+1 for i =1,...,n − 1. In particular, if

θ ∈ SE \{ρe | e ∈ E} 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 120 e f g

h k ℓ k k k ℓe hg kf

Figure 6.1. Boset diagram of E4

then ti = tiαi and αi =1ω(ti) for each i =1,...,n so that

θ = ρt1 ··· ρtn and

ti =6 ti+1 for i =1,...,n − 1. Also, tθ = ∞ and rθ =6 ∞ for all t ∈ T and r ∈ R. 

The proof is by definition of TE, Proposition 4.10 and the observation that

ρrθ,θρr ∈{ρs | s ∈ R}

for all r ∈ R and θ ∈ TE.

It is worth noting that if t, u ∈ T then

Rρt = ω(t) and Rρtρu = ω(t)u.

Also if θ, θ′, θ′′ ∈ T then E b b |Rθ′θθ′′|≤|(Rθ′)θ|≤|Rθ| since Rθ′ ⊆ R.

For examples of egg-box diagrams of TE for sawtooth bosets recall Figure 4.9, Figure 4.12 and Figure 4.15, or see Figure 6.3 in the next section.

6.2. TE for Cyclic Sawtooth Bosets

In this section an infinite class of sawtooth bosets, many of which are nonregular, is intro-

duced for which TE and SE are always regular. Before the definition we motivate the idea

with an example. Let E4 be the sawtooth boset given by the diagram in Figure 6.1. 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 121

Observe that |ω(e)f| = |ω(f)g| = |ω(g)e| =2, |ω(e)g| = |ω(f)e| = |ω(g)f| =1, b b b and b b b x 7→ xg is one-one on ω(e)f = {h, ℓ}, x 7→ xe is one-one on ω(f)g = {k,ℓ}, b x 7→ xf is one-one on ω(g)e = {h, k}. b

This implies the following about products in TE4 : b

If θ ∈ TE4 \{ρr | r ∈ R} then there is a positive integer n and principal ideal isomorphisms

αi : ω(ti) → ω(tiαi) for i =1,...,n with t1,...,tn, t1α1,...,tnαn ∈ T such that

θ = ρα1 ··· ραn and

(tiαi)σ = ti+1 =6 tiαi for i =1,...,n − 1 where σ is the permutation of T given by

σ : e 7→ f, f 7→ g, g 7→ e.

For otherwise

Rρtiαi ρti+1 = ω(e)g, ω(f)e, or ω(g)f so b b b |Rρtiαi ρti+1 | =1

which implies θ = ρr for some r ∈ R. In particular if θ ∈ SE4 \{ρr | r ∈ R} then there is a

positive integer n and t1,...,tn ∈ T such that

θ = ρt1 ··· ρtn and

tiσ = ti+1 =6 ti for i =1,...,n − 1.

It is easy to to check that 2 (ρeρf ρg) ρeρf = ρeρf , 2 (ρf ρgρe) ρf ρg = ρf ρg, 2 (ρgρeρf ) ρgρe = ρgρe,

from which it follows quickly that the egg-box diagram for SE4 is given by the diagram in

Figure 6.2. Some more checking shows that the the egg-box diagram for TE4 is given by the diagram in Figure 6.3. 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 122

ρe ρf ρg

ρeρf ρgρe ρeρf ρeρf ρg

2 2 (ρeρf ρg) ρe (ρeρf ρg) ρeρf ρgρeρf

ρf ρgρe ρf ρgρeρf ρf ρg

2 (ρf ρgρe) ρf ρgρeρf ρg 2 (ρf ρgρe) ρf

ρgρe ρgρeρf ρgρeρf ρg

2 2 ρgρeρf ρgρe (ρgρeρf ) (ρgρeρf ) ρg

ρh ρk ρℓ

Figure 6.2. Egg-box diagram of SE4 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 123

ρe

ρf

ρg ∼ = C2

2 2 (ρeρf ρg) ρe (ρeρf ρg)

2 (ρf ρgρe) 2 (ρf ρgρe) ρf

2 2 ∼ (ρgρeρf ) (ρgρeρf ) ρg = C2

∼ = C1 ρh ρk ρℓ

Figure 6.3. Egg-box diagram of TE4 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 124

We now proceed to the definition. Let E be any sawtooth boset. Call E cyclic if there is a permutation σ of T with finite order such that for all t, u ∈ T there is a positive integer kt such that k if u = tσ |ω(t)u| = t ( 1 if u =6 t, tσ and b x 7→ x(tσ2) is one-one on ω(t)(tσ).

b The following result describes the relationship between the positive integers kt for t ∈ T .

Proposition 6.3. Let E be a cyclic sawtooth boset and t, u ∈ T such that tσn = u for

some positive integer n. Then kt = ku.

Proof. Since the order of σ is finite there is a positive integer m ≥ n such that tσm = t. i Put ti = tσ for i ∈ Z so that

t0 = tm = t, tn = u. For each i ∈ Z we have

kti = |ω(ti)ti+1| = |ω(ti)ti+1ti+2|≤|Rti+1ti+2| = |ω(ti+1)ti+2| = kti+1

since x 7→ xt is one-one on ω(t )t and ω(t ) ⊆ R, so i+2 b b i i+1 i b k ≤ k ≤···≤ k ≤···≤ k = k , t0 b t1 tbn tm t0 which implies

kt0 = kt1 = ··· = ktn = ··· = ktm . In particular

kt = kt0 = ktn = ku. 

By Lemma 3.9 a cyclic sawtooth boset is regular if and only if kt = 1 for all t ∈ T .

We now show that a cyclic sawtooth boset can be constructed for any finite order per- mutation. It is sufficient to show that a cyclic sawtooth boset can be constructed for any cycle, since any finite order permutation σ can be written as a product of disjoint cycles, so that

σ = σi Yi∈I 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 125 where each σi is a cycle and I is an indexing set. Suppose that for each i ∈ I a cyclic sawtooth boset Ei = Ti ⊔ Ri can be constructed for the cycle σi. Define a sawtooth boset E = T ⊔ R where

T = Ti and R = Ri. i∈I i∈I G G Extend the nontrivial multiplication in the following way. For each i, j ∈ I with i =6 j and each u ∈ Tj choose riu ∈ ω(u) and define

ru = riu b for all r ∈ Ri. Then for all t ∈ Ti we have

ω(t)u = {riu} so that b |ω(t)u| =1. It follows quickly that E is a cyclic sawtooth boset for σ. b It remains to show that a cyclic sawtooth boset can be constructed for any cycle σ =

( t1 ··· tn ) of {t1,...,tn} and any positive integer k = kt1 = ··· = ktn . We show this using three cases: n = 1, n = 2 and n ≥ 3. Put T = {t1,...,tn} and let R be a sufficiently large set. Define partial multiplication on E = T ∪ R by (∀r, s ∈ R) rs = s, (∀r ∈ R)(∀t ∈ T ) tr = r.

In each case we choose subsets Si of R which become the sets ω(ti) for i =1,...,n. Note i i that ti = tnσ for each i = 1,...,n. For convenience we use the notation ti = tnσ for all

i ∈ Z so that tn+1 = t1 and t−1 = tn for example. b

Suppose first that n = 1. We must design the boset so that

|ω(t1)| = |ω(t1)t1| = k.

So choose a k-element subset S1 of R and extend the partial multiplication of E by defining b b rt1 for all r ∈ R such that (∀r ∈ S1) rt1 = r,

(∀r ∈ R \ S1) rt1 ∈ S1. Then E is a sawtooth boset with

ω(t1)t1 = ω(t1)= S1 so b b |ω(t1)t1| = k.

Also x 7→ xt1 is one-one on ω(t1)t1. This shows that E is a cyclic sawtooth boset for σ. b b 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 126

For the next two cases we also choose subsets Ui of Si which become the sets ω(ti−1)ti for i =1,...,n. Suppose next n = 2. In the boset we are about to design we must have b ω(t1) ∩ ω(t2) ⊆ ω(t1)t2, ω(t2)t1 so b b b b |ω(t1) ∩ ω(t2)|≤|ω(t1)t2| = |ω(t2)t1| = k.

Choose subsets S1 and S2 of R, each containing at least k-elements, such that the inter- b b b b section S1 ∩ S2 contains at most k elements. Choose k-element subsets U1 of S1 and U2 of

S2 which both contain S1 ∩ S2. Let

α1 : U2 → U1, α2 : U1 → U2

be bijections which fix S1 ∩S2 = U1 ∩U2. Extend the partial multiplication of E by defining rt1 and rt2 for all r ∈ R such that, for i =1, 2 (recalling the convention n +1= 1),

(∀r ∈ Ui) rti = r, rti+1 = rαi+1,

(∀r ∈ Si \ Ui) rti = r, rti+1 ∈ Ui+1,

(∀r ∈ R \ (S1 ∩ S2)) rti ∈ Si.

Any such definition is well defined since both α1 and α2 fix the intersection U1 ∩ U2. It follows quickly that E is a sawtooth boset with

ω(t1)= S1, ω(t2)= S2, ω(t )t = U , ω(t )t = U , b2 1 1 b 1 2 2 so that b b |ω(t1)t2| = |ω(t2)t1| = k.

Also, for each i =1, 2 the mapping x 7→ xti on ω(ti−1)ti is αi which is one-one. This shows E is a cyclic sawtooth boset for σb. b b Suppose finally n ≥ 3. For each i, j =1,...,n with j =6 i, i + 1 we must have

|ω(ti) ∩ ω(tj)|≤|ω(ti)tj| =1.

We choose elements sij ∈ ω(tj) in these cases which become the unique elements of ω(ti)tj. b b b There is a restriction on the choice of sij which will be explained below. We must also avoid the subdiagram in Figureb 6.4 if n ≥ 4, where ti1 =6 ti2 , ti3 , ti2+1, ti3+1, for thenb

|ω(ti2 )ti1 | = |ω(ti3 )ti1 | =1. But then b b r3ti1 = r1 since |ω(ti2 )ti1 | =1 and b r3ti1 = r2 since |ω(ti3 )ti1 | =1

b 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 127

ti1 ti2 ti3

r1 r2 r3

Figure 6.4. Avoidable subdiagram for cyclic sawtooth bosets which is impossible. To avoid this we aim to have

≤ 1 if j = i − 1, i +1 |ω(ti) ∩ ω(tj)| ( =0 if j =6 i − 1, i, i +1 for each i, j =1,...,n. Forb eachbi, j =1,...,n choose subsets Si of R, each containing at least k elements, such that

≤ 1 if j = i − 1, i +1 |Si ∩ Sj| ( =0 if j =6 i − 1, i, i + 1.

Choose k-element subsets Ui of Si which contain the intersections Si−1 ∩ Si. Let

αi : Ui−1 → Ui be a bijection which fixes Si−1 ∩ Si. For each j =1,...,n and each i = j +1,...,j + n − 2 choose elements sij ∈ ω(tj) successively. They must be chosen in order because of the following restriction. If |Sj ∩ Sj+1| = 0 then sj+1,j can be chosen with complete freedom, otherwise sj+1,j is forcedb to be the unique element of Sj ∩ Sj+1. For i = j +2,...,j + n − 2, if |Si−1 ∩ Si| = 0 then si,j can be chosen with complete freedom, otherwise we are forced to have si,j = si−1,j. Extend the partial multiplication of E by defining rt for all r ∈ R and t ∈ T such that

(∀r ∈ Ui) rti = r, rti+1 = rαi+1, rtj = sij if j =6 i, i +1

(∀r ∈ Si \ Ui) rti = r, rti+1 ∈ Ui+1, rtj = sij if j =6 i, i +1

(∀r ∈ R \ (S1 ∪···∪ Sn)) rti ∈ Si.

Again, any such definition is well defined since αi fixes Ui−1 ∩ Ui for each i = 1,...,n. It follows quickly that E is a sawtooth boset with

ω(ti)= Si, ω(ti−1)ti = Ui, ω(ti)tj = {sij} for each i, j =1,...,n with j =6 i, i +1 so that b b b k if j = i +1 |ω(ti)tj| = ( 1 if j =6 i, i +1. b 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 128

Also the mapping x 7→ xti on ω(ti−2)ti−1 = Ui−1 is αi which is one-one for each i =1,...,n. This shows that E is a cyclic sawtooth boset for σ. b When we restrict our attention to cyclic sawtooth bosets Lemma 6.2 becomes the following.

Lemma 6.4. Let E be a cyclic sawtooth boset and

θ ∈ TE \{ρα | α is a principal ideal isomorphism on E}.

Then there exists an integer n ≥ 2 and principal ideal isomorphisms αi : ω(ti) → ω(tiαi)

with ti, tiαi ∈ T for i =1,...,n such that

θ = ρα1 ··· ραn and

(tiαi)σ = ti+1 =6 tiαi for i =1,...,n − 1. In particular, if

θ ∈ SE \{ρe | e ∈ E}

then ti = tiαi and αi =1ω(ti) for each i =1,...,n so that

θ = ρt1 ··· ρtn and

tiσ = ti+1 =6 ti for i =1,...,n − 1. Also, tθ = ∞ and rθ =6 ∞ for all t ∈ T and r ∈ R. 

The proof is essentially the observation that for each i = 1,...,n if ti+1 =6 tiαi, (tiαi)σ then

|Rρtiαi ρti+1 | = |ω(tiαi)ti+1| =1

so θ = ρr for some r ∈ R. b

We now prove regularity of SE.

Theorem 6.5. Let E be any cyclic sawtooth boset. Then SE is regular.

2 Proof. Let θ ∈ SE. If θ = ρe for some e ∈ E then θ = θ is regular. Otherwise

θ ∈ SE \{ρe | e ∈ E} so by Lemma 6.4 there is an integer n ≥ 2 and t1,...,tn ∈ T such that

θ = ρt1 ··· ρtn and

tiσ = ti+1 for i =1,...,n − 1. 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 129

Put tj+1 = tjσ for j ≥ n. Since σ has finite order there is an integer m ≥ n such that tm = t2. Then ρt3 ··· ρtm is one-one on ω(t1)t2 = im(ρt1 ρt2 ) since ρti+2 : x 7→ xti+2 is one-one on ω(ti)ti+1. Since |ω(t1)t2| is finite, ρt3 ··· ρtm is a permutation of ω(t1)t2. So there is a positive integer k such that b

b b k b (ρt3 ··· ρtm |ωb(t1)t2 ) =1ωb(t1)t2 yielding

k−1 k θ(ρtn+1 ··· ρtm ρt3 ··· ρtn ) ρtn+1 ··· ρtm−1 θ = ρt1 ρt2 (ρt3 ··· ρtm ) ρt3 ··· ρtn

= ρt1 ρt2 1ωb(t1)t2 ρt3 ··· ρtn

= ρt1 ··· ρtn = θ. 

In order to prove TE is regular we need the following technical lemma.

Lemma 6.6. Let E be a cyclic sawtooth boset,

θ = ρα1 ··· ραn ∈ TE where n ≥ 2, αi : ω(ti) → ω(tiαi) for i = 1,...,n are principal ideal isomorphisms with t1,...,tn, t1α1,...,tnαn ∈ T and

(tiαi)σ = ti+1 for i =1,...,n − 1, and X a finite subset of R such that θ is one-one on X. Then for each k = 1,...,n − 1 there exists ψk ∈ TE such that

θψk|X = θk|X where θk = ρα1 ··· ραn−k−1 ρtn−k .

Proof. By induction on k. First, suppose k = 1. Since σ has finite order there is a positive integer m such that m tnσ = tn−1αn−1.

So there exists u0,...,um ∈ T such that

u0 = tn,um = tn−1αn−1 and

uiσ = ui+1 for i =0,...,m − 1. Consider the product

−1 −1 −1 θρ ρu1 ··· ρum ρ = θ1ραn−1 ρtn ρu1 ··· ρum ρ . αn αn−1 αn−1 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 130

Observe that ραn−1 ρtn is one-one on Xθ1 since θ = θ1ραn−1 ρtn ραn is one-one on X. Also, by definition of cyclic, ρui is one-one on

Xθ1ραn−1 ρtn ρu1 ··· ρui−1 ⊆ ω(ui−2)ui−1 for i =1,...,m where it is understood that u = u and ρ ··· ρ represents the empty −1 mb u1 u0 string. Finally note that ρ −1 is one-one on αn−1

Xθ1ραn−1 ρtn ρu1 ··· ρum ⊆ ω(tn−1αn−1). These three facts together imply that b

− β = ραn− ρtn ρu ··· ρum ρ 1 1 1 αn−1 is one-one on Xθ1 ⊆ ω(tn−1). Alsob im β ⊆ ω(tn−1), so β|Xθ1 is a partial one-one mapping on ω(tn−1) with finite domain, which can be extended to a principal ideal isomorphism b b b b β : ω(t ) → ω(t ). b n−1 n−1

−1 −1 −1 Put ψ1 = ρ ρu1 ··· ρum ρ ρβ . Then for all x ∈ X, αn αn−1

−1 −1 −1 −1 −1 x(θψ1)= xρα1 ··· ραn ρ ρu1 ··· ρum ρ ρβ =(xθ1)βρβ =(xθ1)ββ = xθ1, αn αn−1 so b θψ1|X = θ1|X which proves the lemma for k = 1. Suppose now that k = ℓ > 1 and, as an inductive hypothesis, the lemma holds for k = ℓ − 1. Then

θψℓ−1|X = θℓ−1|X for some ψℓ−1 ∈ TE by the inductive hypothesis, and

′ θℓ−1ψ1|X = θℓ|X

′ ′ for some ψ1 ∈ TE by the lemma with k = 1. Put ψℓ = ψℓ−1ψ1. Then

′ ′ θψℓ|X = θψℓ−1ψ1|X = θℓ−1ψ1|X = θℓ|X which proves the lemma for k = ℓ. The lemma follows by induction. 

We can now prove that TE is always regular.

Theorem 6.7. Let E be a cyclic sawtooth boset. Then TE is regular.

Proof. Let θ ∈ TE. If θ = ρα for some principal ideal isomorphism α then θ is regular since

ραρα−1 ρα = ραα−1α = ρα. 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 131

Otherwise

θ ∈ TE \{ρα | α is a principal ideal isomorphism on E} so by Lemma 6.4 there is an integer n ≥ 2 and principal ideal isomorphisms

αi : ω(ti) → ω(tiαi) for i =1,...,n with t1,...,tn, t1α1,...,tnαn ∈ T such that

θ = ρα1 ··· ραn and

(tiαi)σ = ti+1 for i =1,...,n − 1.

For each r ∈ Rθ choose xr ∈ ω(t1) such that xrθ = r. Then X = {x | r ∈ Rθ} b r is finite since

Rθ ⊆ (ω(tn−1αn−1)tn)αn and b |ω(tn−1αn−1)tn| < ∞.

By Lemma 6.6 there exists ψ ∈ TE such that b

θψ|X = ρt1 |X . Let r ∈ R. Then

rθψθ = xrθθψθ = xrθρt1 θ = xrθθ = rθ, so θ = θψθ is regular and we are done. 

The following example shows that the definition of cyclic sawtooth bosets cannot be ex-

tended by allowing the associated permutation of T to have infinite order, for then SE and

TE may be nonregular. Define a sawtooth boset E5 in the following way. Let

T = {tn | n ∈ Z}.

For each n ∈ Z define an integer ℓn ≥ 2 by 2n +2 if n ≥ 0 ℓn = ( −2n +1 if n< 0 so that

ℓn = ℓm ⇐⇒ n = m. Put R = ω(t) tG∈T b 6.2. TE FOR CYCLIC SAWTOOTH BOSETS 132

t−1 t0 t1 ··· ···

··· ··· r−1,1 r−1,2 r−1,3 r0,1 r0,2 r1,1 r1,2 r1,3 r1,4

Figure 6.5. Arrow diagram of E5 where

ω(tn)= {rn,1,...,rn,ℓn } for each n ∈ Z. Define partial multiplication on E5 = T ∪ R by b rn,irm,j = rm,j,

rn,itn = rn,i,

rn+1,1 if i =1 rn,itn+1 = ( rn+1,2 if i> 1,

rn,itm = rm,1 if m =6 n, n + 1.

Then E5 is a sawtooth boset with arrow diagram in Figure 6.5. Observe that for each n, m ∈ Z 2 if m = n +1 |ω(tn)tm| = ( 1 if m =6 n, n +1 and the mapping b

ρtn+2 : x 7→ xtn+2 is one-one on

ω(tn)tn+1 = {rn+1,1,rn+1,2}. For each t ∈ T , b ω(t)= {t} ∪ ω(t) so that ∼ b ω(tn) = ω(tm) ⇐⇒ ℓn = |ω(tn)| = |ω(tn)| = ℓm ⇐⇒ tn = tm. Thus all principal ideal isomorphisms are automorphisms of principal ideals. We show that b b ρt0 ρt1 has no inverse in TE5 so that both SE5 and TE5 are not regular. We have

r0,1ρt0 ρt1 = r0,1ρt1 = r1,1,

r0,2ρt0 ρt1 = r0,2ρt1 = r1,2, which implies

ρt0 ρt1 ∈{/ ρr | r ∈ R}. 6.3. REGULARITY OF TE AND SE FOR SOME SAWTOOTH BOSETS 133

We show that

ρt0 ρt1 θρt0 ρt1 ∈{ρr | r ∈ R}

for all θ ∈ TE5 so that

ρt0 ρt1 =6 ρt0 ρt1 θρt0 ρt1

for all θ ∈ TE5 , which shows that ρt0 ρt1 has no inverse in TE5 . If θ = ρr for some r ∈ R then

ρt0 ρt1 θρt0 ρt1 = ρrt0t1 ∈{ρr | r ∈ R}. Otherwise, since all principal ideal isomorphisms are automorphisms, by Lemma 6.4 adapted

to E5, there is a positive integer k, u1,...,uk ∈ T and principal ideal automorphism

αi : ω(ui) → ω(ui) for i =1,...,k

such that

uiσ = ui+1 for i =1,...,k − 1 where σ is permutation of T

σ : tℓ 7→ tℓ+1 for ℓ ∈ Z, and

θ = ρα1 ··· ραk . By Corollary 4.11,

ρt0 ρt1 θρt0 ρt1 = ρt0 ρt1 ρu1 ρα1 ρu1 ··· ρuk ραk ρuk ρt0 ρt1 .

By inspection, ′ ′′ ρt0 ρt1 θρt0 ρt1 = θ ρtn ρtm θ

′ ′′ for some θ , θ ∈ TE and integers m, n such that m < n. But ρtn ρtm = ρrm,1 , so

′ ′′ ρt0 ρt1 θρt0 ρt1 = θ ρrm,1 θ ∈{ρr | r ∈ R}.

6.3. Regularity of TE and SE for some Sawtooth Bosets

In this section we investigate the regularity of TE and SE where E is a sawtooth boset with 2 teeth. The regularity criteria are due to Jordan and appear in [23] Theorem 4.3.6, with some errors. The results and are reproduced here with correct proofs. We start with a result which describes cyclic sawtooth bosets with 2 teeth. 6.3. REGULARITY OF TE AND SE FOR SOME SAWTOOTH BOSETS 134

Lemma 6.8. Let E be a sawtooth boset with T = {t, u} where t =6 u. Then E is cyclic if and only if ω(t)utu = ω(t)u, ω(u)tut = ω(u)t, and both ω(t)u and ω(u)t are finite. b b b b Proof. Firstb supposeb E is cyclic. If the associated permutation σ of T is given by σ : t 7→ t, u 7→ u then |ω(t)u| = |ω(u)t| =1 by definition of cyclic. The set ω(t)utu is a nonempty subset of ω(t)u which contains only one element, so we must have b b b ω(t)utu = ω(t)u. b By symmetry we also have b b ω(u)tut = ω(u)t. Otherwise, σ is given by b b σ : t 7→ u,u 7→ t so

|ω(t)u| = |ω(u)t| = kt < ∞ by definition of cyclic. Also, b b |ω(t)u| = |ω(t)ut| = |ω(t)utu| since x 7→ xt is one-one on ω(t)u and x 7→ xu is one-one on ω(t)ut ⊆ ω(u)t. Since ω(t)utu b b b is a subset of the finite set ω(t)u, we must have b b b b ω(t)utu = ω(t)u. b By symmetry we also have b b ω(u)tut = ω(u)t. Suppose now that ωb(t)utu = ωb(t)u, ω(u)tut = ω(u)t, and both these sets are finite. Then bx 7→ xt is one-oneb on ω(t)u, for otherwise b b |ω(t)utu|≤|ω(t)ut| < |ω(t)u| b so that b b b ω(t)utu ⊂ ω(t)u.

b b 6.3. REGULARITY OF TE AND SE FOR SOME SAWTOOTH BOSETS 135

By symmetry, x 7→ xu is one-one on ω(u)t. This shows that E is cyclic with associated permutation of T given by σb: t 7→ u,u 7→ t. 

We now prove a regularity criterion for SE. The result is due to Jordan and appears in [23] Theorem 4.3.6 (ii). The proof presented here makes use of the theory of cyclic sawtooth bosets.

Theorem 6.9. Let E be a sawtooth boset with T = {t, u} where t =6 u. Then

SE is regular only if ω(t)utu = ω(t)u and ω(u)tut = ω(u)t.

When ω(t)u and ω(u)t are both finite, in particular when E is finite, “only if” becomes “if b b b b and only if”. b b

Proof. Suppose first that SE is regular. Then ρtρu is regular in SE so there exists θ ∈ SE

such that ρtρuθρtρu = ρtρu. If θ = ρr for some r ∈ R then

ρtρu = ρtρuρrρtρu = ρrtu.

Otherwise θ is an alternating product of ρt’s and ρu’s by Lemma 6.2, so

ℓ ρtρu = ρtρuθρtρu = ρtρu(ρtρu) ρtρu for some nonnegative integer ℓ. In either case

k ρtρu =(ρtρu) for some integer k ≥ 2, which implies

k−2 2 2 ω(t)u = Rρtρu =(R(ρtρu) )(ρtρu) ⊆ R(ρtρu) = ω(t)utu ⊆ Rρtρu = ω(t)u, which in turn implies that b b b ω(t)utu = ω(t)u. By symmetry we also have b b ω(u)tut = ω(u)t. Suppose now that b b ω(t)utu = ω(t)u, ω(u)tut = ω(u)t, b b and both of these sets are finite. Then E is cyclic by Lemma 6.8, which implies SE is regular by Theorem 6.5. b b  6.3. REGULARITY OF TE AND SE FOR SOME SAWTOOTH BOSETS 136 t u

··· ··· ··· ···

··· ··· ··· r−1 r0 r1 s−1 s0 s1 k k k k k k

s−2t s−1t s0t r−2u r−1u r0u

Figure 6.6. Boset diagram of E6

The following example shows the finiteness condition in Theorem 6.9 cannot be removed. It is similar to the example used by Jordan in [23] Section 4.3 for the same purpose but the proof here is different. Let E6 be the sawtooth boset with boset diagram in Figure 6.6. where

R = {rn,sn | n ∈ Z}, ω(t)= {rn | n ∈ Z}, ω(u)= {sn | n ∈ Z}, and nontrivial products are given by b b rnu = sn+1, snt = rn+1 For all n ∈ Z. Then ω(t)u = ω(u)= ω(t)utu, ω(u)t = ω(t)= ω(u)tut. b b b We show that ρtρu is not regular in SE6 so that SE6 is not regular. The mapping ρtρu is given by b b b

ρtρu : rn 7→ sn+1, sn 7→ sn+2. By a simple induction, for any positive integer m, we have

m (ρtρu) : rn 7→ sn+2m−1, sn 7→ sn+2m so that m (ρtρu) = ρtρu ⇐⇒ m =1.

Suppose, for a contradiction, that ρtρu is regular in SE5 . Then there exists θ ∈ SE6 such that

ρtρuθρtρu = ρtρu.

If θ = ρr for some r ∈ R then

ρtρu = ρtρuρrρtρu = ρ(rt)u.

Otherwise, by Lemma 6.2, θ is an alternating product of ρt’s and ρu’s, so by Proposition 4.10 there is an integer m ≥ 2 such that

m ρtρu = ρtρuθρtρu =(ρtρu) . 6.3. REGULARITY OF TE AND SE FOR SOME SAWTOOTH BOSETS 137

In either case we have a contradiction.

The next result is due to Jordan, but appears in [23] Lemma 4.3.5 incorrectly without the finiteness condition.

Lemma 6.10. Let E be a sawtooth boset and t, u ∈ T . Then ρtθρu is regular in TE for all

θ ∈ TE if |ω(t)| = |ω(u)| < ∞.

Proof. Let θ ∈ TE and put ψ = ρtθρb u. If ψb= ρα for some principal ideal isomorphism α then ψ is regular by Corollary 4.12. Otherwise im ψ ⊆ R by Lemma 6.2. For each r ∈ im ψ

choose sr ∈ ω(t) such that srψ = r. Define a partial one-one mapping

α : ω(u) → ω(t),r 7→ s . b r Since |ω(t)| = |ω(u)| < ∞, α can be extended to a principal ideal isomorphism b b b α : ω(u) → ω(t). b b b Then for each r ∈ R we have

(rψ)ραψ =(rψ)αψ = srψψ = rψ,

so b ψραψ = ψ,

which shows that ψ = ρtθρu is regular in TE. 

The next result is used implicitly by Jordan in the proof of [23] Theorem 4.3.6. Again, it is used incorrectly without the finiteness condition.

Lemma 6.11. Let E be a sawtooth boset and t, u, v ∈ T such that u =6 v. Then ρtθρuρv is

regular in TE for all θ ∈ TE if

|ω(u)v|≤|ω(v)t| < ∞.

Proof. Let θ ∈ TE and put ψ = ρbtθρuρv. Letb

α : ω(u)v → ω(v)t

be any one-one mapping. For each r ∈ ω(v)t choose s ∈ ω(v) such that s t = r. Define a b b b r r one-one mapping b b β : ω(v)t → ω(v),r 7→ sr.

b b b 6.3. REGULARITY OF TE AND SE FOR SOME SAWTOOTH BOSETS 138

Since ω(u)v is finite, the one-one mapping αβ can be extended to a principal ideal isomor- phism b α : ω(vb)b→ ω(v).

For each x ∈ Rψ choose yx ∈ ω(t) such that yxψ = x. On Rψ ⊆ ω(u)v, ραρt = α is injective, so the mapping b b b γ : ((Rψ)α)t → ω(t), (xα)t 7→ yx is well defined and one-one. Since ω(u)v is finite, γ can be extended to a principal ideal b b isomorphism bγ : ω(t) → ω(tb). For each r ∈ R we have

r(ψραρtργψ) = (((rψ)α)t)γψ = yrψψ = rψ.

This shows that ψραρtργ ψ = ψ, so ψ = ρtθρuρv is regular in TE. 

We can now prove a regularity criterion for TE. The result is due to Jordan, but appears incorrectly, without the finiteness condition, in [23] Theorem 4.3.6 (i). A counterexample for Lemma 6.10, Lemma 6.11 and Theorem 6.12 with the finiteness conditions removed is provided below.

Theorem 6.12. Let E be a sawtooth boset with T = {t, u} where t =6 u. Then

TE is regular only if |ω(t)| = |ω(u)| or |ω(t)u| = |ω(u)t|. When ω(t) and ω(u) are both finite, in particular when E is finite, “only if” becomes “if b b b b and only if”. b b

Proof. First we prove that |ω(t)| = |ω(u)| or |ω(t)u| = |ω(u)t| if TE is regular. We use a

contrapositive argument. Suppose |ω(t)|= 6 |ω(u)| and |ω(t)u|= 6 |ω(u)t|. We show that TE is nonregular. Without loss ofb generalityb we canb assume b b b b b |ω(t)u| > |ω(u)t| > 0.

We show that ρtρu is nonregular in TE. It is sufficient to show that b b |Rρtρuθρtρu|= 6 |Rρtρu|

for all θ ∈ TE for then

Rρtρuθρtρu =6 Rρtρu

for all θ ∈ TE, so ρtρu is nonregular in TE and we will be done. Let θ ∈ TE. If θ = ρr for

some r ∈ R then ρtρuθρtρu = ρrtu, so

|Rρtρuθρtρu| = |{rtu}| =1 < |ω(t)u| = |Rρtρu|.

b 6.3. REGULARITY OF TE AND SE FOR SOME SAWTOOTH BOSETS 139

Otherwise, since ω(t) ≇ ω(u) so that all principal ideal isomorphisms are automorphisms,

and by Lemma 6.2, there is a positive integer n, t1,...,tn ∈ {t, u} and principal ideal automorphisms

αi : ω(ti) → ω(ti) for i =1,...,n such that

θ = ρα1 ··· ραn and

ti =6 ti+1 for i =1,...,n − 1. By Corollary 4.11

ρuθρt = ρuρt1 ρα1 ρt1 ··· ρtn ραn ρtn ρt. By inspection ′ ′′ ρtρuθρtρu = θ ρuρtθ ′ ′′ for some θ , θ ∈ TE which implies

′ ′′ |Rρtρuθρtρu| = |Rθ ρuρtθ |≤|Rρuρt| = |ω(u)t| < |ω(t)u| = |Rρtρu|.

b b Conversely suppose ω(t) and ω(u) are both finite. We prove that TE is regular if |ω(t)| =

|ω(u)| or |ω(t)u| = |ω(u)t|. Let θ ∈ TE. It is sufficient to prove that θ is regular in TE if

either of these conditionsb hold.b If θ = ρα for some principal ideal isomorphism αbthen θ isb regular byb Corollaryb 4.12. Otherwise, by Lemma 6.2 there exists an integer n ≥ 2 and

principal ideal isomorphisms αi : ω(ti) → ω(tiαi) with ti, tiαi ∈{t, u} for i =1,...,n such that

θ = ρα1 ··· ραn and

tiαi =6 ti+1 for i =1,...,n − 1.

By Corollary 4.11, θ = ρt1 θρtnαn . If |ω(t)| = |ω(u)| then θ is regular in TE by Lemma 6.10. Suppose finally |ω(t)u| = |ω(u)t|. Put b b − ψ = θρ 1 = ρ ··· ρ − ρ . b b αn α1 αn 1 tn ′ ′ It is sufficient to prove ψ is regular in TE, for if there exists ψ ∈ TE such that ψψ ψ = ψ then

− ′ ′ θ(ρ 1 ψ )θ = ψψ ψραn = ψραn = θ αn−1

so θ is regular in TE. By Corollary 4.11 we have

ψ = ρt1 (ρα1 ··· ραn−1 )ρtn−1αn−1 ρtn

and tn−1αn−1 =6 tn so ψ is regular in TE by Lemma 6.11 and we are done.  6.3. REGULARITY OF TE AND SE FOR SOME SAWTOOTH BOSETS 140 t u

··· ···

··· ··· r2 r1 r0 s0 s1 s2 k k k k

s1t s0t r0u r1u

Figure 6.7. Boset diagram of E7

The following example shows that the finiteness conditions in Lemma 6.10, Lemma 6.11 or Theorem 6.12 cannot be removed. Let E7 be the sawtooth boset with boset diagram in Figure 6.7. where

R = {rn,sn | n ∈ N}, ω(t)= {rn | n ∈ N}, ω(u)= {sn | n ∈ N}, and nontrivial products are given by b b rnu = sn+1, snt = rn+1 for all n ∈ N. Then |ω(t)| = |N| = |ω(u)|, |ω(t)u| = |N| = |ω(u)t|. b b We show that ρtρu = ρt(ρt)ρtρu is not regular in TE7 so the finiteness conditions cannot b b be removed from Lemma 6.10 or Lemma 6.11. This also shows that TE7 is not regular, so the finiteness condition cannot be removed from Theorem 6.12. Observe that x 7→ xu is one-one on ω(t) and x 7→ xt is one-one on ω(u). Since principal ideal isomorphisms are

also one-one, every element of TE7 \{ρr | r ∈ R} is one-one on both ω(t) and ω(u). The

mapping ρtρbu is given by b

ρtρu : rn 7→ sn+1,sn 7→ sn+2. b b

Suppose, for a contradiction that ρtρu is regular in TE6 . Then there exists θ ∈ TE7 such that

ρtρuθρtρu = ρtρu.

If θ = ρr for some r ∈ R then

ρtρu = ρtρuρrρtρu = ρ(rt)u.

Otherwise θ ∈ TE6 \{ρr | r ∈ R} so θ is one-one on both ω(t) and ω(u). We also have, for each n ∈ N,

sn+2 = snρtρu = ((snρtρu)θ)ρtρu =(bsn+2θ)ρtρbu 6.3. REGULARITY OF TE AND SE FOR SOME SAWTOOTH BOSETS 141 so

sn+2θ = rn+1 for all n ∈ N, or

sn+2θ = sn for all n ∈ N,

since Rθ ⊆ ω(t) or Rθ ⊆ ω(u). If sn+2θ = sn for all n ∈ N then s0θ = sm for some m ∈ N, so

b b s0θ = sm = sm+2θ.

Otherwise sn+2θ = rn+1 for all n ∈ N which implies s0θ = rk and s1θ = rℓ for some k,ℓ ∈ N. Since θ is one-one on ω(u) we must have k =6 ℓ so there exists m ∈ N such that k = m +1 or ℓ = m + 1. This implies that b s0θ = rm+1 = sm+2θ or s1θ = rm+1 = sm+2θ. In any case we have a contradiction.

The last result appears in [23] Corollary 4.3.7 without the finiteness condition, and the proof can be reformulated in terms of cyclic sawtooth bosets.

Corollary 6.13. Let E be a sawtooth boset with T = {t, u} where t =6 u. If SE is regular

and both ω(t)u and ω(u)t are finite, in particular if E is finite, then TE is regular.

Proof. IfbSE is regularb and both ω(t)u and ω(u)t are finite then E is cyclic by Theorem

6.9 and Lemma 6.8, so TE is regular by Theorem 6.7.  b b Bibliography

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SE, 89 FitzGerald idempotent, 10 TE, 28, 80 free semigroup, 12 on a boset, 63 arrow diagram, 58 full transformation, 16 automorphism, 11 full transformation semigroup, 16 Fundamental Homomorphism Theorem, 11 band, 8 fundamental representation boset of an arbitrary semigroup, 70 abstract definition, 52 of an inverse semigroup, 29 of a semigroup, 50 fundamental semigroup, 1, 23 boset arrows, 8, 50 boset diagram, 59 greatest lower bound, 8 Green’s Lemma, 14 Cayley’s Theorem, 16 Green’s relations, 13 coextension, 11 group, 9 biorder-preserving coextension, 68 group inverse, 9 congruence, 10 group presentation, 12 H -congruence, 23 groupoid, 7 biorder-preserving congruence, 68 idempotent separating congruence, 22 homomorphism conjugate, 9 of bosets, 61 cross-connection, 2 of semigroups, 11 cyclic sawtooth boset, 124 idempotent, 8 idempotent-generated semigroup, 11 dual boset, 54 inverse semigroup, 10 dual semigroup, 7 isomorphism duality of bosets, 61 TE, 80 of semigroups, 11 boset, 55 semigroup, 7 Lallement’s Lemma, 23 egg-box, 14 monoid, 9 embedding, 11 Munn semigroup, 28 endomorphism, 11 epimorphism, 11 nontrivial product, 55 144 INDEX 145 partial groupoid, 7 partial transformation, 17 partial transformation semigroup, 17 principal ideal of a boset, 60 of a poset, 28 principal ideal isomorphism of bosets, 62 of posets, 28 regular D-class, 15 regular boset, 64 regular element, 9 regular semigroup, 10 regular-generated semigroup, 11 right zero semigroup, 8 sandwich set, 64 sawtooth boset, 58 semigroup, 7 semigroup inverse, 9 semigroup presentation, 12 semilattice, 8 subboset, 59 subsemigroup, 10 full subsemigroup, 10 symmetric group, 16 symmetric inverse semigroup, 20 trivial product, 55

Wagner-Preston Theorem, 20