On an assertion of J. Rhodes and the finite basis and finite vertex rank problems for pseudovarieties. (Q1421252)

A pseudovariety \(\mathbf U\) of categories is said to have vertex rank \(n\) if a finite category belongs to \(\mathbf U\) whenever all its subcategories with at most \(n\) vertices belong to \(\mathbf U\). The vertex rank of a pseudovariety \(\mathbf V\) of monoids is defined to be the vertex rank of the pseudovariety of categories generated by \(\mathbf V\). A pseudovariety \(\mathbf V\) of monoids is said to be local if it has vertex rank 1. Let \(\mathbf A\) stand for the pseudovariety of all finite aperiodic monoids, \(\mathbf G\) for the pseudovariety of all finite groups, and let \(\overline{\mathbf H}\) be the pseudovariety of all finite monoids whose subgroups belong to a given pseudovariety \(\mathbf H\) of groups. The author shows that if \({\mathbf H}\neq{\mathbf G}\) then every subpseudovariety \(\mathbf V\) of the semidirect product \(\overline{\mathbf H}*{\mathbf G}\) has not finite vertex rank provided that \(\mathbf V\) contains \(\mathbf A\) and a group beyond \(\mathbf H\) (Theorem 2.2). This shows that the vertex rank of the semidirect product of two local pseudovarieties can be infinite; in particular, \({\mathbf A}*{\mathbf G}\) has infinite vertex rank as was asserted by J.~Rhodes. In contrast, the author observes that if \(\mathbf H\) is extension-closed, then the semidirect product \({\mathbf G}*\overline{\mathbf H}\) is local (Theorem 6.5). In particular, \({\mathbf G}*{\mathbf A}\) is local. The proof of Theorem~2.2 is based on a neat construction of a series of finite monoids which is also used to give a new proof to the reviewer's result [Int. J. Algebra Comput. 5, No. 2, 127-135 (1995; Zbl 0834.20058)] that each pseudovariety \(\mathbf V\) satisfying the conditions of the theorem has no finite basis of pseudoidentities.