Almost Factorizable Locally Inverse Semigroups

A factorizable inverse monoid can be identified, up to isomorphism, with an inverse submonoid M of a symmetric inverse monoid I(X) where each element of M is a restriction of a permutation of X belonging to M . So factorizable inverse monoids are natural objects, and appear in a number of branches of mathematics, cf. [12], [4]. The notion of an almost factorizable inverse semigroup was introduced by Lawson [11] (see also [12]) as the semigroup analogue of a factorizable inverse monoid. Among others, he established (see also McAlister [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. Recall that the E-unitary inverse semigroups are just the inverse subsemigroups of semidirect products of semilattices by groups. Thus in the structure theory of inverse semigroups, almost factorizable inverse semigroups have a role dual to that of E-unitary inverse semigroups. The notion of almost factorizability and the basic results mentioned for the inverse case have been generalized in several directions: for straight locally inverse semigroups by Dombi [2], for orthodox semigroups by Hartmann [7] and for right adequate and for weakly ample semigroups by El Qallali [3], and by Gomes and