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.