Local varieties of completely regular monoids
type: Item
instance of: scholarly article
title: Local varieties of completely regular monoids
zbMATH Open document ID: 0774.20036
DOI: 10.1016/S0021-8693(05)80045-6
author: Peter R. Jones
author: Mária B. Szendrei
published in: Journal of Algebra
publication date: 1992-09-27T00:00:00Z
review text: For a variety \(V\) of monoids, let \(\ell V\) be the class of all categories \(C\) whose hom-sets of the form \(C(c,c)\) all belong to \(V\), and \(gV\) the class of all categories which divide a monoid of \(V\). Here division of categories is as defined by \textit{B. Tilson} [J. Pure Appl. Algebra 48, 83-198 (1987; Zbl 0627.20031)]. The variety \(CRn\) of monoids satisfying \(x^{n+1}=x\) consists of monoids which are unions of groups of exponent \(n\). The goal of this paper is to find subvarieties of \(CRn\) which are local, that is, for which \(\ell V=gV\). Let \(S\), \(ReB\), \(RG\) be the varieties of monoid semilattices, rectangular bands and right groups, respectively. For any monoid variety \(S\subseteq V\subseteq CRn\), let \(V_ K\) [resp. \(V_{T_ r}\)] be the smallest among the varieties \(W\) which satisfy \(ReB\circ V=ReB\circ W\) resp. \(RG\circ V=RG\circ W\). Here \(\circ\) denotes the Malcev product. The definition of \(V_{T_ \ell}\) is dual to that of \(V_{T_ r}\). For \(i=0,1\), \(m\geq 0\), define \(V(i,m)\) by \(V(0,0)=V(1,0)=V\), \(V(0,m+1)=V(1,m)_{T_ r}\), \(V(1,m+1)=V(0,m)_{T_ \ell}\). The main theorem of the paper states that for \(S\subseteq V\subseteq CRn\), \(V\) is local if the \(V(i,m)_ K\), whenever nontrivial, are local. If in particular \(V\) consists of orthogroups, that is, if \(V\) satisfies \((x^ ny^ n)^ n=x^ ny^ n\), then the \(V(i,m)_ K\) are group varieties; since nontrivial group varieties are local it thus follows that nontrivial orthogroup varieties are local. A corresponding result holds for pseudovarieties of monoids consisting of orthogroups. The applications form the motivation for the above considerations. Let \(D\) be the pseudovariety of semigroups consisting of the finite semigroups \(S\) such that \(Se=\{e\}\) for each idempotent \(e\) of \(S\). For any nontrivial pseudovariety \(V\) of monoids, let \(LV\) consist of the finite semigroups \(S\) such that \(eSe\in V\) for each idempotent \(e\) of \(S\). Then \(LV\) c…
reviewed by: Francis Pastijn
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M19
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 63890
zbMATH Keywords: variety of monoids
zbMATH Keywords: class of categories
zbMATH Keywords: unions of groups of exponent \(n\)
zbMATH Keywords: hom-sets
zbMATH Keywords: division of categories
zbMATH Keywords: monoid semilattices
zbMATH Keywords: rectangular bands
zbMATH Keywords: right groups
zbMATH Keywords: Malcev product
zbMATH Keywords: orthogroups
zbMATH Keywords: group varieties
zbMATH Keywords: pseudovarieties
zbMATH Keywords: idempotent
zbMATH Keywords: finite orthodox monoids
zbMATH Keywords: bands of idempotents
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 63890