On an assertion of J. Rhodes and the finite basis and finite vertex rank problems for pseudovarieties.
type: Item
instance of: scholarly article
title: On an assertion of J. Rhodes and the finite basis and finite vertex rank problems for pseudovarieties.
zbMATH Open document ID: 1048.20037
DOI: 10.1016/S0022-4049(03)00124-5
author: Benjamin Steinberg
published in: Journal of Pure and Applied Algebra
publication date: 2004-01-26T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 2032645
zbMATH Keywords: monoid pseudovarieties
zbMATH Keywords: semidirect products of pseudovarieties
zbMATH Keywords: finite categories
zbMATH Keywords: pseudovarieties of categories
zbMATH Keywords: finite pseudoidentity bases
zbMATH Keywords: finite vertex rank
zbMATH Keywords: local pseudovarieties
zbMATH Keywords: finite aperiodic monoids
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2032645