Locality of DS and associated varieties (Q1909729)

A variety (or pseudovariety) \(V\) of monoids is local if each (finite) category \(C\) all of whose local monoids are in \(V\) divides a member of \(V\). This concept has been introduced by \textit{B. Tilson} [J. Pure Appl. Algebra 48, 83-198 (1987; Zbl 0627.20031)] and is shown to be an indispensable tool for semidirect decompositions of varieties and pseudovarieties of monoids. In the present paper certain important results concerning locality are proved. First of all, it is shown that for each extension closed pseudovariety \(H\) of groups (meaning that if \(G\) is an extension of \(K\) by \(N\) and \(K,N\in H\) then also \(G \in H\)), the pseudovariety \(DS(H)\), consisting of all finite monoids whose regular \(\mathcal D\)-classes form a subsemigroup and whose subgroups belong to \(H\), is local. It is also shown that for any local pseudovariety \(V\) of completely regular monoids and for each extension closed pseudovariety \(H\) of groups, \(LtHmV\) and \(RHmV\) are again local where \(LtH\) \([RH]\) is the pseudovariety of all left [right] groups with subgroups in \(H\) and \(m\) denotes the Mal'cev product. In particular, the operators \(V \mapsto V^{T_l}\) and \(V \mapsto V^{T_r}\) where \(V^{T_l}=LtGmV=G*V\) and \(V^{T_r}=RGmV=G *_r V\) preserve locality (\(*\) \([*_r]\) denoting the [reverse] semidirect product of pseudovarieties). In the final section of the paper a (pseudo)variety of completely regular monoids is constructed which is not local. More precisely, the [pseudo]variety of (completely regular) monoids defined by the laws \(x^5=x\) and \((xzxwyzyw)^4=(xzxwyzyw)^2\) is shown to be not local.