Rees Matrix Covers for Locally Inverse Semigroups

Abstract

A regular semigroup $S$ is locally inverse if each local submonoid $eSe$, $e$ an idempotent, is an inverse semigroup. It is shown that every locally inverse semigroup is an image of a regular Rees matrix semigroup, over an inverse semigroup, by a homomorphism $\theta$ which is one-to-one on each local submonoid; such a homomorphism is called a local isomorphism. Regular semigroups which are locally isomorphic images of regular Rees matrix semigroups over semilattices are also characterized.