The structure of pseudo-inverse semigroups

Abstract

A regular semigroup S is called a pseudo-inverse semigroup if eSe is an inverse semigroup for each e = e1 G S. We show that every pseudo-inverse semigroup divides a semidirect product of a completely simple semigroup and a semilattice.We thereby give a structure theorem for pseudo-inverse semigroups in terms of groups, semilattices and morphisms.The structure theorem which is presented here generalizes several structure theorems which have been given for particular classes of pseudo-inverse semigroups by several authors, and thus contributes to a unification of the theory.Completely (0-) simple semigroups and inverse semigroups form the first prototypes for the study of pseudo-inverse semigroups.We therefore can say that the theory of regular semigroups began with the study of pseudo-inverse semigroups [40,45]."We may distinguish four successful trends in the papers which since then have dealt with some wider classes of pseudo-inverse semigroups: 1. the subdirect products of completely 0-simple and completely simple semigroups, 2. the generalized inverse semigroups (orthodox pseudo-inverse semigroups, 3. the normal band compositions of inverse semigroups, and 4. Rees matrix semigroups over inverse semigroups (with zero).Subdirect products of completely 0-simple semigroups and completely simple semigroups were initiated in [13, Chapter 2] and studied in great detail in [18] (see also 4 of [14]); this class contains several interesting subclasses: (a) the trees of completely 0-simple semigroups [18] which include the primitive regular semigroups [7, Vol.II,16,39,44,46], (b) the regular locally testable semigroups [50] which include the normal bands [36] and the combinatorial completely 0-simple semigroups, (c) the normal bands of groups [37] which include the semilattices of groups [7, Vol.I], (d) the subdirect products of Brandt semigroups which include the locally testable semigroups which are inverse semigroups [50] and the primitive inverse semigroups [39].The generalized inverse semigroups were introduced in [48] as a special class of orthodox semigroups; they include (a) the inverse semigroups, (b) the orthodox completely 0-simple semigroups [9] and the rectangular groups, (c) the