Local varieties of completely regular monoids
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 \(LV\) coincides with \(V*D\) [\(=V\text{ wr }D\)], the pseudovariety generated by the semidirect [wreath] products \(S*T\) \([S\text{ wr }T]\), \(S\in V\), \(T\in D\). Another consequence of the main theorem states that for \(G\), the pseudovariety of all groups, and \(V\) a pseudovariety of band monoids, \(V*G=\text{V wr }G\) is the monoid pseudovariety generated by the finite orthodox monoids with bands of idempotents in \(V\), or alternatively, is the monoid pseudovariety consisting of the finite monoids whose idempotents form a band which is in \(V\).
scientific article; zbMATH DE number 63890
scientific article