REGULARITY OF TRANSFORMATION SEMIGROUPS DEFINED BY A PARTITION
Pattama Purisang, Jittisak Rakbud · Communications of the Korean Mathematical Society · 2016
Let X be a nonempty set, and let $\mathfrak{F}=\{Y_i:i{\in}I\}$ 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 $i{ eq}j$ . Let ${\emptyset}{ eq}J{\subseteq}I$ , and let $T^{(J)}_{\mathfrak{F}}(X)=\{{\alpha}{\in}T(X):{\forall}i{\in}I{\exists}_j{\in}J,Y_i{\alpha}{\subseteq}Y_j\}$ . Then $T^{(J)}_{\mathfrak{F}}(X)$ is a subsemigroup of the semigroup $T(X,Y^{(J)})$ of functions on X having ranges contained in $Y^{(J)}$ , where $Y^{(J)}:={\bigcup}_{i{\in}J}Y_i$ . For each ${\alpha}{\in}T^{(J)}_{\mathfrak{F}}(X)$ , 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 $T^{(J)}_{\mathfrak{F}}(X)$ as follows: $({\alpha},{\beta}){\in}{\chi}{\Leftrightarrow}{\chi}^{({\alpha})}={\chi}^{({\beta})}$ and $({\alpha},{\beta}){\in}\widetilde{\chi}{\Leftrightarrow}{\chi}^{({\alpha})}{\mid}_J={\chi}^{({\ alpha})}{\mid}_J$ . We begin this paper by studying the regularity of the quotient semigroups $T^{(J)}_{\mathfrak{F}}(X)/{\chi}$ and $T^{(J)}_{\mathfrak{F}}(X)/{\widetilde{\chi}}$ , and the semigroup $T^{(J)}_{\mathfrak{F}}(X)$ . For each ${\alpha}{\in}T_{\mathfrak{F}}(X):=T^{(I)}_{\mathfrak{F}}(X)$ , we see that the equivalence class [ ${\alpha}$ ] of ${\alpha}$ under ${\chi}$ is a subsemigroup of $T_{\mathfrak{F}}(X)$ if and only if ${\chi}^{({\alpha})}$ is an idempotent element in the full transformation semigroup T(I). Let $I_{\mathfrak{F}}(X)$ , $S_{\mathfrak{F}}(X)$ and $B_{\mathfrak{F}}(X)$ be the sets of functions in $T_{\mathfrak{F}}(X)$ such that ${\chi}^{({\alpha})}$ is injective, surjective and bijective respectively. We end this paper by investigating the regularity of the subsemigroups [ ${\alpha}$ ], $I_{\mathfrak{F}}(X)$ , $S_{\mathfrak{F}}(X)$ and $B_{\mathfrak{F}}(X)$ of $T_{\mathfrak{F}}(X)$ .