Regularity and green's relations on semigroups of full transformations with surjective restriction on the fixed set

Abstract

Let X be a nonempty set and T(X) denote the semigroup (under composition) of transformations from X to itself and let S(X, Y) = {α ∈ T(X) I Y α ⊆ Y } where Y is a nonempty subset of X. Then S (X, Y) is a semigroup of total transformations on X which leave a subset Y of X invariant. In this thesis, we study the subsemigroup of S(X, Y) defined by T(X,Y ) = {α ∈ T(X) | Y α = Y } Here, we characterize the regular elements of T(x,y) and give necessary and sufficient conditions for T(x,y) to be a regular semigroup. Then we describe Green's relations on T(xy). Finally, we characterize when T(x,y) is isomorphic to T(Z) for some set Z and prove that every semigroup A can be embedded in T(A0,1,{0}) when A0,1 = (A ∪ {0})1