Quasi-ideal embeddings and Rees matrix covers for regular semigroups
type: Item
instance of: scholarly article
title: Quasi-ideal embeddings and Rees matrix covers for regular semigroups
zbMATH Open document ID: 0770.20026
DOI: 10.1016/0021-8693(92)90094-3
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: Let \(\mathcal C\) be a class of semigroups. We say that a semigroup \(S\) is a locally \(\mathcal C\)-semigroup if \(eSe \in {\mathcal C}\) for every idempotent \(e \in S\). A semigroup \(S\) is called a locally isomorphic image of a regular Rees matrix semigroup over \(\mathcal C\) if \(S\) is an image under a homomorphism from a regular Rees matrix semigroup over \(\mathcal C\) which is one-to-one on \(eSe\) for every idempotent \(e \in S\). A subsemigroup \(Q\) of \(S\) is said to be a quasi-ideal of \(S\) if \(QSQ \subseteq S\). Let \(\mathcal C\) be one of the following classes: all semilattices, all inverse semigroups, all \({\mathcal L}\)-unipotent semigroups, all orthodox semigroups, all unions of groups, or all \(E\)-solid semigroups. Then a locally \(\mathcal C\)-semigroup \(S\) is a locally isomorphic image of a regular Rees matrix semigroup over \(\mathcal C\) if and only if \(S\) can be embedded as a quasi- ideal into a locally \(\mathcal C\)-semigroup with greatest \(\mathcal J\)-class. Moreover, if \(S\) is a locally inverse semigroup, or a locally \(\mathcal L\)- unipotent semigroup, or a locally orthodox semigroup, or a locally \(E\)- solid semigroup then both conditions hold. If \(S\) is a locally testable semigroup, or a locally union of groups then both conditions hold if and only if the principal ideals of \(S\) form a semilattice under intersection.
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 130503
zbMATH Keywords: locally \(\mathcal C\)-semigroup
zbMATH Keywords: idempotent
zbMATH Keywords: locally isomorphic image
zbMATH Keywords: regular Rees matrix semigroup
zbMATH Keywords: semilattices
zbMATH Keywords: \({\mathcal L}\)-unipotent semigroups
zbMATH Keywords: orthodox semigroups
zbMATH Keywords: unions of groups
zbMATH Keywords: quasi-ideal
zbMATH Keywords: principal ideals
author: Donald B. McAlister
reviewed by: Václav Koubek
MaRDI profile type: Publication
full work available at URL: 0021-8693(92)90094-3
OpenAlex ID: W1966168922
cites work: A generalization of the Rees theorem to a class of regular semigroups
cites work: Q3929783
cites work: Rees Matrix Covers for Locally Inverse Semigroups
cites work: Rees matrix covers for regular semigroups
cites work: Q3689376
cites work: Q3699026
cites work: The natural partial order on a regular semigroup
description: scientific article; zbMATH DE number 130503