Almost factorisable inverse semigroups

Abstract

0. Introduction. In [3], McAlister introduced a class of semigroups, called covering semigroups, which were shown to play an important role in the theory of E-unitary covers of semigroups. Strangely, this class of semigroups appears to have received little attention subsequently. It is the aim of this paper to rehabilitate them and to study their properties in more detail. As a first step, we have chosen to rename them almost factorisable semigroups, since they can be regarded as the semigroup analogues of factorisable inverse monoids. Before discussing the contents of this paper in more detail we recall some standard terminology.We assume the reader is familiar with basic inverse semigroup theory. For convenience, however, we describe here the notation and terminology used which is not explicitly defined in the text. All semigroups will be inverse. The (generalised) inverse of an element x will be denoted by x~\ The set of idempotents of a semigroup is written £(S); if / I c 5 then E(A) = A (1 E(S). The natural partial order is denoted by « . A semigroup 5 is E-unitary if e ^x and e e E(S) implies x e £(5). An E-unitary cover of an inverse semigroup S is an E-unitary semigroup P and a surjective, idempotent separating homomorphism d:P-*S. It is well-known that every inverse semigroup has an E-unitary cover [7]. If 5 is a monoid then the group of units of 5 will be denoted by U(S). An inverse monoid S is said to be factorisable if for each s eS there exists g e U(S) such that s ^g. The minimum group congruence o on a semigroup S is denned by xoy if, and only if, there exists z e S such that z < x and z^y. If (Q, £ ) is a poset then A c Q is said to be an order ideal if b < a eA implies that b eA. The set [a] = {b e Q:b <«} is called a principal order ideal. The trace product of x and y in a semigroup 5 is defined to be xy if x~lx =yy~l, and undefined otherwise. If A and B are sets then the projections from A x B to A and B respectively are denoted by n x and jt 2 . If 6:A^B is a function then ker 6 denotes the equivalence relation induced on A. Finally, if p is a congruence then p* denotes the corresponding natural map.The paper is divided into three sections. In Section 1, we discuss the basic properties of almost factorisable inverse semigroups and show that they are very closely related to factorisable inverse monoids. Specifically, we prove in Proposition 1.7 that an inverse monoid is almost factorisable if, and only if, it is factorisable. In Theorem 1.10, we show that for every almost factorisable inverse semigroup S there is a factorisable inverse monoid F such that S is isomorphic to F\U(F). In Section 2, we describe all strongly E-unitary covers of almost factorisable inverse semigroups. This is achieved in Proposition 2.3 and Theorem 2.5. In Theorem 2.4, we show that the class of almost factorisable semigroups is the closure under homomorphisms of the class of strongly E-unitary semigroups. In the final section, Section 3, we relate arbitrary inverse semigroups to almost factorisable inverse semigroups...

Enlargements of regular semigroups

“…Example 2. In [18], we showed that an E-unitary inverse semigroup is over a semilattice (in the sense of McAlister [25]) if, and only if, it has an enlargement which is a semidirect product of a group and a semilattice. On the basis of this result, we proved in [18] that every inverse semigroup has an enlargement which is close to being factorisable (what we call almost factorisable), sharpening the results of [24] and [29].…”

Almost Factorizable Locally Inverse Semigroups

“…Now we establish that, for inverse semigroups, almost factorizability in our sense coincides with the usual notion of almost factorizability (see [11,12]). Conversely, assume that S is an inverse semigroup and U is a completely simple subsemigroup in O(S) within properties (AF1) and (AF2).…”

“…The notion of an almost factorizable inverse semigroup was introduced by Lawson [11] (see also [12]) as the semigroup analog of a factorizable inverse monoid. Among others, he established (see also [13] where the main ideas and some of the results were implicit) that the almost factorizable inverse semigroups are just the homomorphic images (or, equivalently, the idempotent separating homomorphic images) of semidirect products of semilattices by groups.…”

Factorizable inverse monoids

“…11.1; [32], section 8.1) and latterly the re ‡ection monoids of Everitt and Fountain [8]. Appropriate generalisations of the concept were developed: signi…cantly, Lawson [17] identi…ed the appropriate generalisation from monoids to semigroups as almost factorizable semigroups, which had been used in McAlister [25]; for an account, see section 7.1 of [18]. Tirasupa [35] examined the Cli¤ord by semilattice case and Mills [29] the group by aperiodic case.…”