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 \[T_{E}(X)=\{\alpha\in T(X) : \forall a, b\in X, (a, b)\in E \Rightarrow (a\alpha, b\alpha\in E)\}.\] 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) $.