Groups, Semilattices and Inverse Semigroups

Regular semigroups, fundamental semigroups and groups

“…An inverse semigroup S is called £-unitary if the equations ea = e = e 2 in S together imply that a 2 = e. It is shown in McAlister (1974), that every inverse semigroup S has an £-unitary cover P in the sense of the following lemma which summarizes the results on E-unitary inverse semigroups which we shall need in this paper. LEMMA 0.…”

Algebraic properties of Zappa–Szép products of semigroups and monoids

“…One of the fundamental results in the structure theory of semigroups is that every inverse semigroup is an idempotent separating homomorphic image of a 'proper' inverse semigroup where a proper inverse semigroup is isomorphic to a so called 'P-semigroup' and embeds into a semidirect product of a semilattice by a group [17][18][19]. For (left) restriction semigroups, many authors have produced work using constructions similar (at least on the surface) to those of McAlister, replacing groups by monoids of various kinds.…”

Factorization Theorems for Morphisms of Ordered Groupoids and Inverse Semigroups

“…Lawson has shown [8] that this theorem lies behind the classical theory of idempotent-pure homomorphisms and prehomomorphisms of inverse semigroups due to O'Carroll [16,17] and McAlister [12] (see [11,Chapter 8,§ 4]). Lawson showed that such a K has a universal property which characterizes it up to isomorphism as the unique such enlargement [11,Theorem 8.3.5].…”

“…He then proves, with an entirely coordinate-free approach, the McAlister P -theorem [12], and redevelops O'Carroll's theory of idempotent-pure homomorphisms of inverse semigroups [16,17] using the techniques of ordered groupoids. The key to Lawson's approach is Ehresmann's Maximum Enlargement Theorem [11,Theorem 8.3.3], characterizing star-injective morphisms of ordered groupoids.…”