Strong Morita equivalence of semigroups with local units.
type: Item
instance of: scholarly article
title: Strong Morita equivalence of semigroups with local units.
zbMATH Open document ID: 1237.20057
author: Valdis Laan
author: L. Márki
published in: Journal of Pure and Applied Algebra
publication date: 2011-08-19T00:00:00Z
review text: This paper is based on Morita contexts to develop Morita equivalence for semigroups. Let \(S\) and \(T\) be semigroups, \(E(S)\) be the set of idempotent elements in \(S\). A semigroup with local units is a semigroup \(S\) such that for any \(s\in S\) there exist \(e,f\in E(S)\) such that \(s=esf\). A left \(S\)-act \(_SA\) (a biact \(_SA_T\)) is said to be unitary if \(SA=A\) (\(SA=AT=A\)). A semigroup \(S\) is factorisable if the \(S\)-act \(_SS\) is unitary. A unitary Morita context is a six-tuple \((S,T,{_SP_T},{_TQ_S},\theta,\varphi)\), where \(\theta\colon{_S(P\otimes_TQ)_S}\to{_SS_S}\) and \(\varphi\colon{_T(Q\otimes_SP)_T}\to{_TT_T}\) are biact morphisms such that for any \(p,p'\in P\) and \(q,q'\in Q\), \(\theta(p\otimes q)p'=p\varphi(q\otimes p')\) and \(q\theta(p\otimes q')=\varphi(q\otimes p)p'\). Semigroups \(S\) and \(T\) are called strongly Morita equivalent if there exists a unitary Morita context such that the mappings \(\varphi\) and \(\theta\) are surjective. For a factorisable semigroup \(S\), it is proved that a Rees matrix semigroup \(\mathcal M=\mathcal M(S,U,V,M)\) is strongly Morita equivalent to \(S\) if and only if \(S\text{im}(M)S=S\), where \(\text{im}(M)\subseteq S\) is the set of the entries of the sandwich matrix \(M\). It is proved that if \(S\) and \(T\) are strongly Morita equivalent semigroups then there exists a Rees matrix semigroup \(\mathcal M=\mathcal M(S,U,V,M)\) and an epimorphism \(\tau\colon\mathcal M\to T\) such that the restriction of \(\tau\) to every subsemigroup \(a\mathcal Mb\) of \(\mathcal M\) is injective, where \(a\in\mathcal Ma\) and \(b\in b\mathcal M\) (such an epimorphism is called a strict local isomorphism), and the idempotents and regular elements lift along \(\tau\), that means for any idempotent \(f\in T\) (resp. any regular element \(t\in T\)) there exists an idempotent \(e\in\mathcal M\) (resp. a regular element \(s\in\mathcal M\)) such that \(\tau(e)=f\) (resp. \(\tau(t)=s\)). Some elegant characteri…
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 5941012
zbMATH Keywords: semigroups with local units
zbMATH Keywords: acts
zbMATH Keywords: Morita equivalences
zbMATH Keywords: Rees matrix semigroups
zbMATH Keywords: Morita semigroups
zbMATH Keywords: idempotents
zbMATH Keywords: biacts
zbMATH Keywords: unitary Morita contexts
reviewed by: Chen Yuqun
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.02.017
OpenAlex ID: W2152661286
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DOI: 10.1016/J.JPAA.2011.02.017
description: scientific article; zbMATH DE number 5941012