The semidirect product of an inverse semigroup and a group

Abstract

It is shown that a semidirect product of an inverse semigroup and a group, in that order, contains an inverse subsemigroup that is a retract and that, together with the retraction mapping, forms a free inverse morphic image of the semidirect product. The congruence determined by the retraction mapping is shown to be determined by the semigroup of idempotents of the semidirect product.