### Green's Relations and Regularity for the Self-E-Preserving Transformation Semigroups

#### Abstract

Let $T_X$ be the full transformation semigroup on a set $X$ and $E$ an arbitrary equivalence relation on $X$. We define a subsemigroup of $T_X$ as follows: \[ T_{SE}(X) = \{ \alpha \in T_X : \forall x\in X, (x, x\alpha) \in E \} \]which is called the \textit{self-E-preserving transformation semigroup} on $X$. Then $T_{SE}(X)$ becomes a regular semigroup. The purpose of this paper is to investigate Green's relations for $T_{SE}(X)$. Moreover, we characterize when certain elements of $T_{SE}(X)$ are left regular, right regular and completely regular.

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