Idempotent pure extensions by inverse semigroups via quivers

Abstract

In this paper we describe idempotent pure regular extensions by inverse semigroups by means of quivers and actions of inverse semigroups, generalising the category and action of groups approach presented by Margolis and Pin for E-unitary inverse semigroups. By making use of these new tools, we can uniformly reprove O'Carroll's and Billhardt's characterisations of idempotent pure inverse extensions by inverse semigroups as L m -semigroups and as inverse subsemigroups of a λ-semidirect product of a semilattice by an inverse semigroup, respectively.