Context equivalence of semigroups.
type: Item
instance of: scholarly article
title: Context equivalence of semigroups.
zbMATH Open document ID: 1214.20061
author: Valdis Laan
published in: Periodica Mathematica Hungarica
publication date: 2011-04-26T00:00:00Z
review text: Let \(S\) and \(T\) be arbitrary semigroups. A Morita context is a six-tuple \((S,T,{_SP_T},{_TQ_S},\theta,\varphi)\), where \(_SP_T\) is an \((S,T)\)-bi-act, \(_TQ_S\) is a \((T,S)\)-bi-act, and \(\theta\colon{_S(P\otimes_TQ)_S}\to{_SS_S}\), \(\varphi\colon{_T(Q\otimes_SP)_T}\to{_TT_T}\) are bi-act morphisms such that, for every \(p,p'\in P\) and \(q,q'\in Q\), \(\theta(p\otimes q)p'=p\varphi(q\otimes p')\), \(q\theta(p\otimes q')=\varphi(q\otimes p)q'\). Semigroups \(S\) and \(T\) are called strongly Morita equivalent if there exists a Morita context such that \(_SP_T\) and \(_TQ_S\) are unitary bi-acts and the mappings \(\theta\) and \(\varphi\) are surjective. Semigroups \(S\) and \(T\) are called context equivalent if there exists a Morita context such that the mappings \(\theta\) and \(\varphi\) are surjective. It is proved that context equivalence is an equivalence relation on the class of factorizable semigroups and that Cauchy completions of context equivalent semigroups are equivalent categories. Among other results, the following is proved: a semigroup \(S\) is strongly Morita equivalent (or, equivalently, factorizable and context equivalent) 1) to a semigroup of local units if and only if \(S\) is a sandwich semigroup; 2) to a monoid if and only if \(S=SeS\) for some \(e\in E\); 3) to a group if and only if \(S\) is completely simple; 4) to a one-element semigroup if and only if \(S\) is a rectangular band.
reviewed by: Peeter Normak
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 5881211
zbMATH Keywords: bi-acts
zbMATH Keywords: strong Morita equivalences
zbMATH Keywords: Morita contexts
zbMATH Keywords: context equivalences
zbMATH Keywords: factorisable semigroups
zbMATH Keywords: inverse semigroups
zbMATH Keywords: semilattices
MaRDI profile type: Publication
full work available at URL: s10998-010-1081-z
OpenAlex ID: W2031077307
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DOI: 10.1007/S10998-010-1081-Z
description: scientific article; zbMATH DE number 5881211