Groups, semilattices and inverse semigroups. II

Abstract

An inverse semigroup is called proper if the equations a e = e = e 2 ae = e = {e^2} together imply a 2 = a {a^2} = a . In a previous paper, with the same title, the author proved that every inverse semigroup is an idempotent separating homomorphic image of a proper inverse semigroup. In this paper a structure theorem is given for all proper inverse semigroups in terms of partially ordered sets and groups acting on them by order automorphisms. As a consequence of these two theorems, and Preston’s construction for idempotent separating congruences on inverse semigroups, one can give a structure theorem for all inverse semigroups in terms of groups and partially ordered sets.