Embedding theorems for proper inverse semigroups

Abstract

Let S be an inverse semigroup with semilattice of idempotents E, and let ϱ( S), or ϱ if there is no danger of ambiguity, be the minimum group congruence on S. Then S is said to be proper if Eϱ = E, or alternatively, if the equation ex = e for some e ϵ E and x ϵ S implies that x ϵ E. For example, free inverse semigroups and fundamental ω-inverse semigroups are proper. In a recent paper, McAlister has given a remarkable structure theorem for an arbitrary proper inverse semigroup, and using this theorem we show (Theorem 1.3) that any proper inverse semigroup P can be embedded in a semidirect product P ̄ of a semilattice and a group. Some consequences of this result are given; for example, if P is bisimple with identity then P ̄ is simple (see Theorem 1.6). Reilly has proved that an arbitrary inverse semigroup can be embedded in a bisimple inverse semigroup with identity. Given a proper inverse semigroup P it is shown that a proper inverse semigroup P′, arising from P by blowing up P g9 , can indeed be embedded in a bisimple proper inverse semigroup with identity (Theorem 2.4). We also investigate those homomorphic images of bisimple proper inverse semigroups which are themselves proper. In the final section, Rees quotient images of proper inverse semigroups are characterised (Theorem 3.4), and some other constructions for (simple) semidirect products of semilattices and groups are considered.