Regularity and Green's Relations on a Semigroup of Transformations with Restricted Range

Let T(X) be the full transformation semigroup on the set X and let T(X,Y)={α∈T(X):Xα⊆Y}. Then T(X,Y) is a sub-semigroup of T(X) determined by a nonempty subset Y of X. In this paper, we give a necessary and sufficient condition for T(X,Y) to be regular. In the case that T(X,Y) is not regular, the largest regular sub-semigroup is obtained and this sub-semigroup is shown to determine the Green's relations on T(X,Y). Also, a class of maximal inverse sub-semigroups of T(X,Y) is obtained.