On a class of left type-\(A\) monoids
type: Item
instance of: scholarly article
title: On a class of left type-\(A\) monoids
zbMATH Open document ID: 0957.20035
author: Jianfei Chen
author: Changxiang Rui
published in: Semigroup Forum
publication date: 2001-03-20T00:00:00Z
review text: For a semigroup \(S\), let \({\mathcal R}^*\) be the equivalence relation on \(S\) defined by: \(a{\mathcal R}^*b\) if and only if \(a\) and \(b\) are \(\mathcal R\)-related in some oversemigroup of \(S\). A monoid \(S\) is called a left type-\(A\) monoid if (i) for every \(a\in S\), the \({\mathcal R}^*\)-class of \(a\) contains a unique idempotent \(a^+\), (ii) the set of idempotents of \(S\) forms a semilattice, and, (iii) \(ae=(ae)^+a\) for all \(a\in S\) and every idempotent \(e\) of \(S\). Let \(S\) be a left type-\(A\) monoid, \(\tau\) the congruence relation on \(S\) generated by \({\mathcal R}^*\). It is shown that there exists a congruence relation \(\rho\) on \(S\) such that \(\rho\) induces the least right cancellative congruence on each of the \(\tau\)-classes. The left type-\(A\) monoid is called weakly proper if \(\rho\cap{\mathcal R}^*\) is the equality on \(S\). The authors prove several results which serve as a motivation for the study of this class of monoids and they give a structure theorem for these weakly proper left type-\(A\) monoids.
reviewed by: Francis Pastijn
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 1423175
zbMATH Keywords: right cancellative monoids
zbMATH Keywords: \(E\)-reflexive semigroups
zbMATH Keywords: equivalences
zbMATH Keywords: idempotents
zbMATH Keywords: congruence relations
zbMATH Keywords: right cancellative congruences
zbMATH Keywords: weakly proper left type-\(A\) monoids
MaRDI profile type: Publication
full work available at URL: s002339910021
OpenAlex ID: W2035355331
DOI: 10.1007/S002339910021
description: scientific article; zbMATH DE number 1423175