The regular part of a semigroup of transformations with restricted range

Abstract

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed subset Y of X. It is known that F(X, Y)={αT(X, Y): Xα⊆ Y=α}, is the largest regular subsemigroup of T(X,Y) and determines Green's relations on T(X,Y). In this paper, we show that F(X,Y)≅T(Z) if and only if X=Y and {pipe}Y{pipe}={pipe}Z{pipe}; or {pipe}Y{pipe}=1={pipe}Z{pipe}, and prove that every regular semigroup S can be embedded in F(S1,S). Then we describe Green's relations and ideals of F(X,Y) and apply these results to get all of its maximal regular subsemigroups when Y is a nonempty finite subset of X. © 2011 Springer Science+Business Media, LLC.