Right Magnifying Elements in E-preserving Transformation Semigroups with Restricted Range

Thananya Kaewnoi, Montakarn Petapirak, Ronnason Chinram

Abstract

An element $a$ of  a semigroup $S$ is called a right magnifying \mbox{element} if  there exists a proper subset $M$ of $S$  such that $S=Ma$.Given $Y$ as a nonempty subset of a set $X,$ the subsemigroup of the full transformation semigroup $T(X)$ on $X,$ consisting of all transformations on $X$ whose range is contained in $Y,$ is denoted by $T(X,Y).$ Let $E$ be an equivalence relation on $X$ and $T_E(X,Y)=\{\alpha\in T(X,Y)\mid (x,y)\in E$ implies $((x)\alpha,(y)\alpha)\in E\}.$ In this paper, we provide the necessary and sufficient conditions for elements in $T_E(X,Y)$ to be right magnifying.