Green's relations and regularity on some subsemigroups of transformations that preserve equivalences

Authors: NARES SAWATRAKSA, CHAIWAT NAMNAK

Abstract: Let $T(X)$ be the full transformation semigroup on a set $X$. For two equivalence relations $E$ and $F$ on $X$ with $F \subseteq E$, let

$T(X, E, F) = \{ \alpha \in T(X) : \forall x, y \in X, (x, y)\in E \Rightarrow (x\alpha, y\alpha) \in F \}. $

Then $T(X, E, F)$ is a subsemigroup of $T(X)$. In this paper, we describe Green's relations and the regularity of elements for $T(X, E, F)$. Also, the relations $F$ and $E$ for which $T(X, E, F)$ is a regular semigroup are described.

Keywords: Transformation semigroup, equivalence, GreenÕs relations, regular.

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