Semigroups of Full Transformations with the Restriction on the Fixed Set is Bijective

Ekkachai Laysirikul

Abstract

Let $T(X)$ be the full transformation semigroup of the set $X$ and let $S(X, Y)=\{\alpha \in T(X) : Y\alpha \subseteq Y\}$ where $Y$ is a nonempty subset of $X$. Then $S(X, Y)$ is a subsemigroup of $T(X)$ which is not regular. In this paper, let \[ PG_Y(X)=\{\alpha \in T(X) : \alpha|_Y \in G(Y) \} \] where $G(Y)$ is the permutation group on $Y$. Then $PG_Y(X)$ is a subsemigroup of $S(X, Y)$. Some relationships between $PG_Y(X)$ it's subsemigroup and $S(X, Y)$ are considered. Moreover, it is shown that $PG_Y(X)$ is regular and characterizations of left re