Regularity on Variants of Transformation Semigroups that Preserve an Equivalence Relation

Abstract

The variant of a semigroup $S$ with respect to an element $a\in S$, is the semigroup with underlying set $S$ and a new binary operation $\ast$ defined by $x\ast y=xay$ for $x,y\in S$. Let $T(X)$ be the full transformation semigroup on a nonempty set $X$. For an arbitrary equivalence $E$ on $X$, let $$T_E(X)=\{\alpha \in T(X):\mathrm{\forall }a,b\in X,(a,b)\in E\Rightarrow (a\alpha ,b\alpha \in E)\}.$$
Then $T_E(X)$ is a subsemigroup of $T(X)$. In this paper, we investigate regular, left regular and right regular elements for the variant of some subsemigroups of the semigroup $T_E(X)$.