Transformation Semigroups Which Are Disjoint Union of Symmetric Groups

Abstract: Let $X$ be a nonempty set and $T(X)$ the full transformation semigroup on $X$. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by $$ T_{E^*}(X)={\alpha\in T(X):\text{for all}\ x,y\in X, (x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E}. $$ We have the regular part of $T_{E^*}(X)$, denoted by $\mathrm{Reg}(T)$, is the largest regular subsemigroup of $T_{E^*}(X)$. Defined the subsemigroup $Q_{E^*}(X)$ of $T_{E^*}(X)$ by $$ Q_{E^*}(X)={\alpha\in T_{E^*}(X):|A\alpha|=1\ \text{and}\ A\cap X\alpha\neq\emptyset\ \text{for all}\ A\in X/E}. $$ Then we can prove that this subsemigroup is the (unique) minimal ideal of $\mathrm{Reg}(T)$ which is called the kernel of $\mathrm{Reg}(T)$. In this paper, we will compute the rank of $Q_{E^*}(X)$ when $X$ is finite and prove an isomorphism theorem. Finally, we describe and count all maximal subsemigroups of $Q_{E^*}(X)$ where $X$ is a finite set.