Communications of the
Korean Mathematical Society CKMS

Let $X$ be a nonempty set,and let $\mathscr{F}=\{Y_{i}:i\in I\}$ be a family of nonempty subsets of $X$ with the properties that $X=\bigcup_{i\in I}{}Y_{i}$, and $Y_{i}\cap Y_{j}=\emptyset$ for all $i,j\in I$ with $i\neq j$. Let $\emptyset\neq J\subseteq I$, and let $T_{\mathscr{F}}^{(J)}(X)=\left\{\alpha\in T(X):\forall i\in I\exists j\in J, Y_{i}\alpha\subseteq Y_{j}\right\}$. Then $T_{\mathscr{F}}^{(J)}(X)$ is a subsemigroup of the semigroup $T\left(X,Y^{(J)}\right)$ of functions on $X$ having ranges contained in $Y^{(J)}$, where $Y^{(J)}:=\bigcup_{i\in J}{}Y_{i}$. For each $\alpha\in T_{\mathscr{F}}^{(J)}(X)$, let $\chi^{(\alpha)}:I\rightarrow J$ be defined by $i\chi^{(\alpha)}=j\Leftrightarrow Y_{i}\alpha\subseteq Y_{j}$. Next, we define two congruence relations $\chi$ and $\widetilde{\chi}$ on $T_{\mathscr{F}}^{(J)}(X)$ as follows: $(\alpha,\beta)\in\chi\Leftrightarrow \chi^{(\alpha)}=\chi^{(\beta)}$ and $(\alpha,\beta)\in\widetilde{\chi}\Leftrightarrow \chi^{(\alpha)}|_{J}=\chi^{(\beta)}|_{J}$. We begin this paper by studying the regularity of the quotient semigroups $T_{\mathscr{F}}^{(J)}(X)/\chi$ and $T_{\mathscr{F}}^{(J)}(X)/\widetilde{\chi}$, and the semigroup $T_{\mathscr{F}}^{(J)}(X)$. For each $\alpha\in T_{\mathscr{F}}(X):=T_{\mathscr{F}}^{(I)}(X)$, we see that the equivalence class $[\alpha]$ of $\alpha$ under $\chi$ is a subsemigroup of $T_{\mathscr{F}}(X)$ if and only if $\chi^{(\alpha)}$ is an idempotent element in the full transformation semigroup $T(I)$. Let $I_{\mathscr{F}}(X)$, $S_{\mathscr{F}}(X)$ and $B_{\mathscr{F}}(X)$ be the sets of functions $\alpha$ in $T_{\mathscr{F}}(X)$ such that $\chi^{(\alpha)}$ is injective, surjective and bijective respectively. We end this paper by investigating the regularity of the subsemigroups $[\alpha]$, $I_{\mathscr{F}}(X)$, $S_{\mathscr{F}}(X)$ and $B_{\mathscr{F}}(X)$ of $T_{\mathscr{F}}(X)$.

Keywords: full transformation semigroup, regular element, character

MSC numbers: Primary 20M17; Secondary 20M20