The natural partial order on the semigroup of all transformations of a set that reflect an equivalence relation

Authors

  • L. Sun
  • X. Xin

Keywords:

transformation semigroup, natural partial order, compatibility, maximal(minimal) elements

Abstract

Let ${\cal T}_X$ be the full transformation semigroup on a set $X$ and $E$ be a non-trivial equivalence relation on $X$. Denote $$T_\exists (X) =\{ f\in {\cal T}_X : \forall \, x,y\in X,\, (f(x),f(y))\in E \Rightarrow (x,y)\in E\},$$ then $T_\exists (X) $ is a subsemigroup of ${\cal 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 characterize the maximal elements and the minimal elements. 10.1017/S0004972712001013

Published

2013-09-27

Issue

Vol. 88 No. 3 (2013)

Section

Articles
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