Semigroups of transformations with invariant set

Abstract

Let T(X) denote the semigroup (under composition) of trans-formations from X into itself. For a xed nonempty subset Y of X, let S(X, Y) = {α ε T(X): Y α ⊆ Y} Then S(X, Y) is a semigroup of total transformations of X which leave a subset Y of X invariant. In this paper, we characterize when S(X, Y) is isomorphic to T(Z) for some set Z and prove that every semigroup A can be embedded in S(A1;A). Then we describe Green's relations for S(X, Y) and apply these results to obtain its group H-classes and ideals. © 2011 The Korean Mathematical Society.