Some results on semigroups of transformations with restricted range

Let X X be a non-empty set and Y Y a non-empty subset of X X . Denote the full transformation semigroup on X X by T ( X ) T\left(X) and write f ( X ) = { f ( x ) ∣ x ∈ X } f\left(X)=\{f\left(x)| x\in X\} for each f ∈ T ( X ) f\in T\left(X) . It is well known that T ( X , Y ) = { f ∈ T ( X ) ∣ f ( X ) ⊆ Y } T\left(X,Y)=\{f\in T\left(X)| f\left(X)\subseteq Y\} is a subsemigroup of T ( X ) T\left(X) and R T ( X , Y ) RT\left(X,Y) , the set of all regular elements of T ( X , Y ) T\left(X,Y) , also forms a subsemigroup of T ( X , Y ) T\left(X,Y) . Green’s ∗ \ast -relations and Green’s ∼ \sim \hspace{0.08em} -relations (with respect to a non-empty subset U U of the set of idempotents) were introduced by Fountain in 1979 and Lawson in 1991, respectively. In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup T ( X , Y ) T\left(X,Y) . This investigation proves that the semigroup T ( X , Y ) T\left(X,Y) is always a left Ehresmann semigroup, and R T ( X , Y ) RT\left(X,Y) is orthodox (resp. completely regular) if and only if Y Y contains at most two elements.