SEMIGROUPS OF TRANSFORMATIONS WITH FIXED SETS

摘要

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 (0)≠Y(∈)X. However, its rank is related to the rank of T(XY) and the cardinality of Y when X is a finite set.