Semigroup of transformations with restricted partial range: regularity, abundance and some combinatorial results

Suppose that X be a nonempty set. Denote by T (X) the full transformation semigroup on X. For empty set =? Z subset of Y subset of X, let T (X, Y, Z) = {alpha is an element of T (X) : Y alpha subset of Z}. Then, T (X, Y, Z) is a subsemigroup of T (X). In this paper, we characterize the regular elements of the semigroup T (X, Y, Z), and present a necessary and sufficient condition under which T (X, Y, Z) is regular. Furthermore, we investigate the abundance of the semigroup T (X, Y, Z) for the case Z ? Y ? X. In addition, we compute the cardinalities of T (X, Y, Z), Reg(T (X, Y, Z)) and E(T (X, Y, Z)) when X is finite, respectively.