Abundance of $E$-order-preserving transformation semigroups

Authors: LEI SUN, XUEFENG HAN

Abstract: Let ${\cal T}_X$ be the full transformation semigroup on a finite totally ordered set $X=\{1<2<\ldots<n\}\,(n\geq3)$ and $E$ be a nontrivial equivalence relation on $X$. In this paper, we consider a subsemigroup of ${\cal T}_X$ defined by $$EOP_X=\{f\in {\cal T}_X:\forall\,x,y\in X,\,(x,y)\in E,x\leq y\Rightarrow (f(x),f(y))\in E,f(x)\leq f(y)\}$$ and present a necessary and sufficient condition under which the semigroup $EOP_X$ is abundant.

Keywords: Transformation semigroup, ${\cal L}^*$-relation, ${\cal R}^*$-relation, idempotent, abundance

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