Congruences and group congruences on a semigroup.
type: Item
instance of: scholarly article
title: Congruences and group congruences on a semigroup.
zbMATH Open document ID: 1281.20070
author: Roman S. Gigoń
published in: Semigroup Forum
publication date: 2013-06-18T00:00:00Z
review text: Let \(S\) be a semigroup and \(E\) be the set of idempotents of \(S\). A subsemigroup \(N\) of \(S\) is called normal if it is full (i.e., \(E\subseteq S\)), dense (i.e., \((\forall s\in S)(\exists x,y\in S)(sx,ys\in N)\)), reflexive (i.e., \((\forall a,b\in S)(ab\in N\Rightarrow ba\in N)\)) and closed in \(S\) (i.e., \(N\omega=N\), where \(N\omega=\{s\in S:(\exists a\in N)(as\in N)\)). A semigroup \(S\) is said to be \(E\)-inversive if for every \(a\in S\) there exists \(x\in S\) such that \(ax\in E\). \(S\) is an \(E\)-semigroup if \(E\) is a subsemigroup of \(S\). Regular \(E\)-semigroups are called orthodox. A semigroup with \(E\neq\emptyset\) is said to be \(E\)-unitary if \(E\) is unitary subset of \(S\), i.e. \(se\in E\) implies \(s\in E\) and \(es\in E\) implies \(s\in E\) for every \(s\in S\), \(e\in E\). The author shows that there is an inclusion-preserving bijection between the set of all normal subsemigroups of a semigroup \(S\) and the set of all group congruences on \(S\). Also group congruences on \(E\)-inversive (\(E\)-)semigroups are described. In particular, these results generalize the result of \textit{J. Meakin} [J. Aust. Math. Soc. 13, 259-266 (1972; Zbl 0265.20056)] concerning the description of the least group congruence on an orthodox semigroup, the result of \textit{J. M. Howie} [Proc. Edinb. Math. Soc., II. Ser. 14, 71-79 (1964; Zbl 0123.01701)] concerning the description of \(\rho\vee\sigma\) in an inverse semigroup \(S\), where \(\rho\) is a congruence and \(\sigma\) is the least group congruence on \(S\), and some other known results. Also, one of the main aims of the reviewed paper is to study group congruences on \(E\)-unitary semigroups. In particular, it is proved that in any \(E\)-inversive semigroup, \(\mathcal H\cap\sigma\subseteq\kappa\), where \(\kappa\) is the least \(E\)-unitary congruence. This result is equivalent to the statement that in an arbitrary \(E\)-unitary \(E\)-inversive semigroup \(S\), \(\mathcal H\cap\sigma=1_…
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 6176653
zbMATH Keywords: group congruences
zbMATH Keywords: \(E\)-inversive semigroups
zbMATH Keywords: \(E\)-semigroups
zbMATH Keywords: idempotents
zbMATH Keywords: idempotent-surjective semigroups
zbMATH Keywords: eventually regular semigroups
zbMATH Keywords: idempotent pure congruences
zbMATH Keywords: idempotent-separating congruences
zbMATH Keywords: \(E\)-unitary congruences
reviewed by: Leonid M. Martynov
MaRDI profile type: Publication
full work available at URL: s00233-012-9425-z
OpenAlex ID: W2132311513
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DOI: 10.1007/S00233-012-9425-Z
description: scientific article; zbMATH DE number 6176653