Semilattice decompositions of semigroups revisited
type: Item
instance of: scholarly article
title: Semilattice decompositions of semigroups revisited
zbMATH Open document ID: 0968.20030
author: Stojan Bogdanović
author: Miroslav Ćirić
author: Žarko Popović
published in: Semigroup Forum
publication date: 2001-06-18T00:00:00Z
review text: Let \(N\) denote the set of all positive integers. Let \(S\) be a semigroup, and \(a,b\in S\). Define relations \(\mid\), \(\to\), and \(-\), respectively by \(a\mid b\Leftrightarrow (\exists x,y\in S^1) b=xay\), \(a\to b\Leftrightarrow (\exists n\in N) a\mid b^n\), \(a-b\Leftrightarrow a\to b\to a\). We associate a digraph \((S,\to)\) and a graph \((S,-)\) corresponding to \(\to\) and \(-\), respectively. If there exists a path from \(a\) to \(b\) in \((S,\to)\) (respectively, a path between \(a\) and \(b\) in \((S,-)\)) then there exist paths from \(a\) to \(b\) (respectively between \(a\) and \(b\)) of minimal length. Those are called the minimal paths from \(a\) to \(b\) (respectively between \(a\) and \(b\)). The rank of \(S\), \(\text{ran}(S)\), is the supremum of the lengths of all minimal paths in \((S,-)\), and the semirank of \(S\), \(\text{sran}(S)\), is the supremum of the lengths of all minimal paths in \((S,\to)\). The \(n\)-th powers of \(\to\) and \(-\) are, respectively, denoted by \(\to^n\) and \(-^n\). Then \(\text{ran}(S)\) is the smallest \(n\leq\infty\) for which \(-^n\) is transitive, and \(\text{sran}(S)\) is the smallest \(n\leq\infty\) for which \(\rightarrow^n\) is transitive. Next for \(n\in N\) and \(a\in S\) define the subsets \(\Sigma_n(a)\) and \(\Sigma(a)\) of \(S\) as follows: \(\Sigma_n(a)=\{x\in S\mid a\to^n x\}\), \(\Sigma(a)=\{x\in S\mid a\to^\infty x\}\) and define \(\sigma_n\) and \(\sigma\) on \(S\): \((a,b)\in\sigma_n\Leftrightarrow\Sigma_n(a)=\Sigma_n(b)\), \((a,b)\in\sigma\Leftrightarrow\Sigma(a)=\Sigma(b)\). Further, let \(\widehat\Sigma_n(a)=\{x\in S\mid a-^nx\}\), \((a,b)\in\widehat\sigma_n\Leftrightarrow\widehat\Sigma_n(a)=\widehat\Sigma_n(b)\). According to Ćirić and Bogdanović, \(\sigma\) is the smallest semilattice congruence on \(S\). Then \(S\) is \(\sigma_n\)-simple if and only if \(a-^nb\) for all \(a,b\in S\). Also \(\widehat\sigma_n\)-simpleness implies \(\sigma_n\)-simpleness. In this paper the structure of \…
reviewed by: Takayuki Tamura
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 1523492
zbMATH Keywords: digraphs
zbMATH Keywords: graphs
zbMATH Keywords: minimal paths
zbMATH Keywords: semilattice congruences
zbMATH Keywords: band congruences
zbMATH Keywords: semilattices of Archimedean semigroups
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1523492