Magnifying elements of semigroups of transformations with invariant set

Let T(X) be a semigroup of all transformations on a set X. Let Y be a fixed nonempty subset of X and S(X, Y) = {f is an element of T(X)vertical bar(Y)f subset of Y}. Then S(X, Y) is a subsemigroup of T(X) which leaves a subset Y of X invariant. If Y = X, then S(X, Y) = S(X, X) = T(X). In this paper, we give necessary and sufficient conditions for elements in S(X, Y) to be left or right magnifying.