### Green’s Relations and Regularity for Semigroups of Transformations with Restricted Range that Preserve Double Direction Equivalence Relations

#### Abstract

Let $T(X)$ be the full transformation semigroup on a set $X$. For an equivalence $E$ on $X$ and a nonempty subset $Y$ of $X$, let $$T_{E^*}(X,Y)=\{ \alpha\in T(X):X\alpha\subseteq Y\ \text{and}\ \forall x,y\in X,(x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E\}.$$ In this article, we give a necessary and sufficient condition for $T_{E^*}(X,Y)$ to be a subsemigroup of $T(X)$ under the composition of functions and study the regularity of $T_{E^*}(X,Y)$. Finally, we characterize Green's relations on this semigroup.

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