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THE NATURAL PARTIAL ORDER ON THE SEMIGROUP OF ALL TRANSFORMATIONS OF A SET THAT REFLECT AN EQUIVALENCE RELATION

Part of: Semigroups

Published online by Cambridge University Press:  25 January 2013

LEI SUN  and
XIANGJUN XIN
LEI SUN*
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Henan, Jiaozuo 454003, PR China
XIANGJUN XIN
Affiliation:
Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Henan, Zhengzhou 450002, PR China email xin_xiang_jun@126.com
*
✉sunlei97@163.com
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Abstract

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Let ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote

$$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$
so that ${T}_{\exists } (X)$ is a subsemigroup of ${ \mathcal{T} }_{X} $. In this paper, we endow ${T}_{\exists } (X)$ with the natural partial order and investigate when two elements are related, then find elements which are compatible. Also, we characterise the minimal and maximal elements.

Keywords

transformation semigroupnatural partial ordercompatibilityminimal (maximal) elements

MSC classification

Secondary: 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 359 - 368
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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