The Idempotent-separating Congruences on a Regular 0-bisimple Semigroup

Abstract

A congruence ρ on a semigroup is said to be idempotent-separating if each ρ-class contains at most one idempotent. For any idempotent e of a semigroup S the set eSe is a subsemigroup of S with identity e and group of units He, the maximal subgroup of S containing e. The purpose of the present note is to show that if S is a regular O-bisimple semigroup and e is a non-zero idempotent of 5 then there is a one-to-one correspondence between the idempotentseparating congruences on 5 and the subgroups N of He with the property that aN ⊆ Na for all right units a of eSe and Nb ⊆ bN for all left units b of eSe. Some special cases of this result are discussed and, in the final section, an application is made to the principal factors of the full transformation semigroup on a set X.