Regular Elements of the Variant Semigroups of Transformations Preserving Double Direction Equivalences
Winita Yonthanthum · Thai Journal of Mathematics · 2018
Denote $T(X)$ the full transformation semigroup on aset $X$. For an equivalence relation $E$ on $X$, let\[T_{E^*}(X) = \{\alpha \in T(X) \mid \forall x, y \in X, (x, y) \in E \Leftrightarrow (x\alpha, y\alpha) \in E\}.\]Then $T_{E^*}(X)$ is a subsemigroup of $T(X)$. For $\theta \in T_{E^*}(X)$, we define a sandwich operation $\ast$ on$T_{E^*}(X)$ by $\alpha \ast \beta = \alpha\theta\beta$ where$\alpha\theta\beta$ is the composition of functions $\alpha,\theta$ and $\beta$. Under this operation, $T_{E^*}(X)$ is asemigroup which is called the variant semigroup of $T_{E^*}(X)$ with the sandwich function $\theta$, and denotedby $(T_{E^*}(X), \theta)$. In this paper, we give a necessary andsufficient condition for an element of $(T_{E^*}(X), \theta)$ tobe regular and consider when $(T_{E^*}(X), \theta)$ is a regular semigroup.