Regularity of some invariant transformation semigroups

Let Y be a nonempty subset of X. The invariant subsemigroup of the full transformation semigroup T(X) is denoted by T-Y(X) = {alpha is an element of T(X) : Y alpha subset of Y}. In this paper, we characterize the regularity of the following subsemigroups: AM(Y)(X) = {alpha. T-Y(X) : {x is an element of X : | x alpha alpha(-1)| > 1} is finite}, AE(Y)(X) = {alpha T-Y(X) : X\ ran alpha is finite} and the complement sets of both AM(Y)(X) and AE(Y)(X). These are generalizations of certain subsemigroups of T(X). Furthermore, infinitely many regular and nonregular elements of each semigroup are given.