Note on Transitive Representations of Generalized Inverse *-Semigroups (Languages, Algebra and Computer Systems)
Isamu Inata, 輝男 今岡 · Institutional Repositories DataBase (IRDB) · 1999
In [1], we obtained that an effective representation of a locally [generalized] $\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}*$ -semigroup $S$ is the sum of a uniquely determined family of transitive representations of $S$ .In this paper, we will determine a transitive represen- tation of a generalized inverse $*$ -semigroup by using right $\omega$ -cosets.This is a generalization of Schein's result [5] for inverse semigroups. 1 Introduction*}=e$ .Denote the sets of idempotents and projections of $S$ by $E(S)$ and $P(S)$ , respectively.Let $S$ be a $\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}*$ -semigroup.If $eSe$ is an inverse semigroup, for every $e\in E(S)$ , $S$ is called a locally $inverse*$ -semigroup.If $E(S)$ is a normal band, that is, it satisfies the identity $xyzx=$ xzyx, $S$ is called a generalized inverse $*$ -semigroup.A regular $*$ -semigroup $S$ is a generalized $\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}*$ -semigroup if and only if it is a locally inverse $*$ -semigroup and $E(S)$ forms aFor a subset A of a $\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}*$ -semigroup $S$ , the set $A\omega--$ { $x\in S$ : there exists $a\in A$ such that $a\leq x$ } is called the closure of $A$ .The following statements are easily verified.