Semigroups of transformations with fixed sets

Abstract

Let T (X) denote the semigroup (under composition) of transformations from X into itself. For a fixed subset Y of X, let Fix(X, Y) be the set of all self-maps on X which fix all elements in Y. Then Fix(X, Y) is a regular subsemigroup of T (X). The aim of this paper is to determine the Green's relations and ideals of Fix(X, Y) and prove that Fix(X, Y) is never isomorphic to T (Z) for any set Z when O = Y X. However, its rank is related to the rank of T (X \Y) and the cardinality of Y when X is a finite set.