Abundance and magnifying elements in semigroups of transformations with restricted range that preserve an equivalence

Let ρ be an equivalence relation on a set X and T(X,ρ) be the semigroup of all transformations of X that preserve ρ. For a subset Y of X, let TY(X,ρ)={α∈T(X,ρ):Xα⊆Y}, then TY(X,ρ) is a subsemigroup of T(X,ρ). In this paper, we give a necessary and sufficient condition for TY(X,ρ) to be abundant. Also, we describe all the magnifying elements in TY(X,ρ).