# Regularity on Variants of Transformation Semigroups that Preserve an Equivalence Relation

## Authors

• Piyaporn Tantong Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University
• Nares Sawatraksa

## Keywords:

Transformation semigroup, Equivalence relation, Left regular, Right regular, Completely regular

## 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 $T_{E}(X)$.

2022-10-31

## Section

Nonlinear Analysis

## How to Cite

Regularity on Variants of Transformation Semigroups that Preserve an Equivalence Relation. (2022). European Journal of Pure and Applied Mathematics, 15(4), 2116-2126. https://doi.org/10.29020/nybg.ejpam.v15i4.4596

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