Natural partial order in semigroups of transformations with invariant set

Authors

  • Lei Sun
  • I Wang

Abstract

Let ${\cal T}_X$ be the full transformation semigroup on the nonempty set $X$. We fix a nonempty subset $Y$ of $X$ and consider the semigroup $$S(X,Y) =\{f\in {\cal T}_X :f(Y)\subseteq Y\}$$ of transformations that leave $Y$ of $X$ invariant and endow it with the so-called natural partial order. Under this partial order, we determine when two elements of $S(X,Y)$ are related, find the elements which are compatible and describe the maximal elements, the minimal elements and the greatest lower bound of two elements. Also, we show that the semigroup $S(X,Y)$ is abundant. DOI: 10.1017/S0004972712000287

Published

2012-12-17

Issue

Section

Articles