Regularity and abundance on semigroups of transformations preserving an equivalence relation on an invariant set

Let $T(X)$ be the full transformation semigroup on a nonempty set $X$. For an equivalence relation $E$ on $X$ and a nonempty subset $Y$ of $X$, let

$${\overline{S}}_E(X,Y)=\{\alpha \in T(X):\mathrm{\forall }x,y\in Y,(x,y)\in E\Rightarrow (x\alpha ,y\alpha )\in E,x\alpha ,y\alpha \in Y\}.$$

Then ${\overline{S}}_E(X,Y)$ is a subsemigroup of $T(X)$ consisting of all full transformations that leave $Y$ and the equivalence relation $E$ on $Y$ invariant. In this paper, we show that ${\overline{S}}_E(X,Y)$ is not regular in general and determine all its regular elements. Then we characterize relations $\mathcal{L}$, ${\mathcal{L}}^{\ast }$, $\mathcal{R}$ and ${\mathcal{R}}^{\ast }$ on ${\overline{S}}_E(X,Y)$ and apply these characterizations to obtain the abundance on such semigroup.