Beyond orthodox semigroups.
type: Item
instance of: scholarly article
title: Beyond orthodox semigroups.
zbMATH Open document ID: 1275.20067
author: Victoria Gould
author: Yanhui Wang
published in: Journal of Algebra
publication date: 2013-04-17T00:00:00Z
review text: Let \(S\) be a semigroup and \(\emptyset\neq B\subseteq E(S)\). Let \(\widetilde{\mathcal L}_B\) be an equivalence relation, so that for \(a,b\in S\), \(a\widetilde{\mathcal L}_Bb\) if and only if \(\{e\in B:ae=a\}=\{e\in B:be=b\}\). A semigroup \(S\) is said to be \textit{weakly B-abundant} if every \(\widetilde{\mathcal L}_B\)-class and every \(\widetilde{\mathcal R}_B\)-class contains an idempotent of \(B\). If \(\widetilde{\mathcal L}_B\) is a right congruence and \(\widetilde{\mathcal R}_B\) is a left congruence, then \(S\) is said to be satisfying the \textit{Congruence Condition} (C). A weakly \(B\)-abundant semigroup is said to be \textit{weakly B-orthodox} if it has (C) and \(B\) is a band. The authors define a \textit{generalised category} and an \textit{inductive generalised category} (over a band \(B\)) and prove that the category of weakly \(B\)-orthodox semigroups and admissible morphisms is isomorphic to the category of inductive generalised categories over bands and pseudo-functors. Special cases of the categories of orthodox and some other semigroups are discussed as well.
reviewed by: Peeter Normak
Mathematics Subject Classification ID: 20M19
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 18A05
zbMATH DE Number: 6154887
zbMATH Keywords: orthodox semigroups
zbMATH Keywords: inductive generalised categories
zbMATH Keywords: generalised groupoids
zbMATH Keywords: abundant semigroups
zbMATH Keywords: idempotents
zbMATH Keywords: Green relations
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.06.012
OpenAlex ID: W1964245224
cites work: Structure of concordant semigroups
cites work: The structure of type A semigroups
cites work: THE FREE AMPLE MONOID
cites work: Left adequate and left Ehresmann monoids. II.
cites work: FREE ADEQUATE SEMIGROUPS
cites work: Retracts of trees and free left adequate semigroups
cites work: Semigroups and ordered categories. I: The reduced case
cites work: On the structure of inverse semigroups
cites work: The structure mappings on a regular semigroup
cites work: Q3889272
cites work: Structure of regular semigroups. I: Fundamental regular semigroups
cites work: Structure of regular semigroups. II: The general case
cites work: On \(U\)-orthodox semigroups.
cites work: Weakly \(B\)-orthodox semigroups.
DOI: 10.1016/J.JALGEBRA.2012.06.012
description: scientific article; zbMATH DE number 6154887