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 TuUy 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.

Free Objects in Joins of Strict Inverse and Completely Simple Semigroup Varieties

“…On the other hand, Tully [26] was the first to introduce a similar construction as above (with a special semilattice X and each I a being 0-simple) when he studied semigroups for which each ideal is a retract. Generalizing the results of Tully, the author [1] has shown that the class of all globally idempotent semigroups all of whose principal ideals being a retract is characterized by the construction S = (X;I a ,f a p ), X being a semilattice and each I a being 0-simple. The above mentioned construction of strict regular semigroups is useful for several purposes.…”

“…Next we give an interpretation of the ^-classes of F{3S v s/ ) r DEFINITION. Let / be a non-empty set and 5 be a semigroup with a unary operation xt-^x' 1 The main result of the present section is the following. THEOREM 3.9.…”