Structure of concordant semigroups
type: Item
instance of: scholarly article
title: Structure of concordant semigroups
zbMATH Open document ID: 0653.20064
DOI: 10.1016/0021-8693(88)90057-9
author: Sheena Armstrong
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: The relations \({\mathcal L}^*\) and \({\mathcal R}^*\) in a semigroup S are defined as follows: elements \(a,b\in S\) are related by \({\mathcal L}^*\) (\({\mathcal R}^*)\) iff they are related by \({\mathcal L}\) (\({\mathcal R})\) in some oversemigroup of S. A semigroup is abundant if each \({\mathcal L}^*\)-class and each \({\mathcal R}^*\)-class contains an idempotent. Abundant semigroups are thus generalizations of regular semigroups. Parts of the theory of regular semigroups have been generalized to abundant semigroups in a series of papers by J. B. Fountain and his pupils. The present paper also belongs to this line. A concordant semigroup is an abundant semigroup in which the idempotents generate a regular subsemigroup, and which also satisfies a condition called idempotent-connectedness, relating the idempotents in the \({\mathcal L}^*\)-class of an element to those in the \({\mathcal R}^*\)-class of the element. E.g., full subsemigroups of regular semigroups or Rees matrix semigroups over a cancellative monoid with the sandwich matrix entries coming from the group of units, are concordant semigroups. The paper contains two main results. The first of them determines the structure of concordant semigroups in terms of inductive cancellative categories, and shows in fact that the categories of concordant semigroups and of inductive cancellative categories are equivalent. This generalizes the corresponding results of \textit{K. S. S. Nambooripad} [Mem. Am. Math. Soc. 224 (1979; Zbl 0457.20051)] on regular semigroups. (A cancellative category is a small category with left and right cancellation for morphisms; inductiveness is defined by conditions which in the groupoid case reduce to those used by Nambooripad.) The inductive cancellative category associated with a concordant semigroup S consists of, as morphisms, the triples \((e,x,f)\in S^ 3\), where e and f are idempotents belonging to the \({\mathcal R}^*\)-class and \({\mathcal L}^*\)- class, respectively, …
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 18B40
zbMATH DE Number: 4065308
zbMATH Keywords: \({\mathcal L}^*\)-class
zbMATH Keywords: \({\mathcal R}^*\)-class
zbMATH Keywords: regular semigroups
zbMATH Keywords: abundant semigroups
zbMATH Keywords: idempotents
zbMATH Keywords: idempotent-connectedness
zbMATH Keywords: concordant semigroups
zbMATH Keywords: inductive cancellative categories
zbMATH Keywords: categories of concordant semigroups
zbMATH Keywords: category with left and right cancellation for morphisms
zbMATH Keywords: Lipschitz condition
zbMATH Keywords: policy functions
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90057-9
OpenAlex ID: W2049360145
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description: scientific article; zbMATH DE number 4065308