The Maximum Idempotent-Separating Congruence on an Inverse Semigroup

Abstract

A congruence ρ on a semigroup will be called idempotent-separating if each ρ-class contains at most one idempotent. It is shown below that there exists a maximum such congruence µ on every inverse semigroup S. Two characterisations of µ are found, and it is shown (a) that S/µ⋍E, the semilattice of idempotents of S, if and only if E is contained in the centre of S; (b) that µ is the identical congruence on S if and only if E is self-centralising, in a sense explained below.