REGULARITY OF TRANSFORMATION SEMIGROUPS DEFINED BY A PARTITION

Title & Authors
REGULARITY OF TRANSFORMATION SEMIGROUPS DEFINED BY A PARTITION
Purisang, Pattama; Rakbud, Jittisak;

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