Left regular and right regular elements of the semigroups of transformations restricted by an equivalence

Ekkachai Laysirikul; Nares Sawatraksa; Chaiwat Namnak

doi:10.14456/nujst.2018.25

Authors

Ekkachai Laysirikul Department of Mathematics, Faculty of Science, Naresuan University
Nares Sawatraksa nares_math@hotmail.com
Chaiwat Namnak Department of Mathematics, Faculty of Science, Naresuan University

DOI:

https://doi.org/10.14456/nujst.2018.25

Abstract

Let $gif.latex?T(X)$ be the semigroup of all transformations on a set $gif.latex?X$ . For an arbitrary equivalence relation $gif.latex?\sigma$ on $gif.latex?X$ , we consider a subset of $gif.latex?T(X)$ defined by

$gif.latex?E(x,\sigma&space;)=\{\alpha&space;\in&space;T(X)&space;:&space;\forall&space;x,y&space;\in&space;X&space;\Rightarrow&space;x\alpha=y\alpha\}.$

It is obvious that $gif.latex?E(X,\sigma)$ is a subsemigroup of $gif.latex?T(X)$ . In this paper, we characterize the left regular, the right regular and the completely regular for elements of $gif.latex?E(X,\sigma)$ . Moreover, we give a necessary and sufficient conditions for the semigroup $gif.latex?E(X,\sigma)$ when it is left regular, right regular and completely regular, respectively.

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