Generalised retract semigroups

Abstract

We give a description of the structure of the semigroups for which each principal ideal is a retract. The globally idempotent case is solved quickly using a—suitably modified—construction which has been developed by Tully for the study of semigroups in which each ideal is a retract. The general case can be treated by a naturally obtained semigroup of subsets of the semigroup constructed for the globally idempotent case.