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

Let T X be the full transformation semigroup on the setX 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.