Free spectra of nilpotent varieties
type: Item
instance of: scholarly article
title: Free spectra of nilpotent varieties
zbMATH Open document ID: 0647.08005
DOI: 10.1007/BF01195266
published in: Algebra Universalis
publication date: 1987-01-01T00:00:00Z
review text: The free spectrum of a locally finite variety \({\mathcal V}\) is the sequence \(<| F_{{\mathcal V}}(n)| >\) where \(F_{{\mathcal V}}(n)\) denotes the free n-generator \({\mathcal V}\)-algebra. \textit{P. M. Neumann} [Q. J. Math., Oxf., II. Ser. 14, 46-50 (1963; Zbl 0114.019)] and \textit{G. Higman} [Proc. Int. Conf. Theory of Groups, Canberra 1965, 153-165 (1967; Zbl 0166.280)] showed that, when the algebras concerned are groups, then the function \(\log | F_{{\mathcal V}}(n)|\) is a polynomial in n if and only if \({\mathcal V}\) is nilpotent. This leads naturally to the question as to whether a similar result holds for more general types of algebras where a concept of nilpotence is definable. The first theorem proved is: Let \({\mathcal V}\) be a finitely generated congruence uniform variety. Then either \(\log | F_{{\mathcal V}}(n)|\) is a polynomial in n, or there is a polynomial c(n) such that, for some positive constant C, \(\log | F_{{\mathcal V}}(n)| \geq C\cdot (2^ n-c(n))\) for all n. An example is then given of a nilpotent loop in which the second situation arises. However a second theorem shows that a positive answer can be obtained when the variety in question is generated by a special type of nilpotent algebra.
Mathematics Subject Classification ID: 08B20
zbMATH DE Number: 4055693
zbMATH Keywords: free spectrum of a locally finite variety
zbMATH Keywords: nilpotence
zbMATH Keywords: finitely generated congruence uniform variety
zbMATH Keywords: nilpotent loop
zbMATH Keywords: nilpotent algebra
author: Joel Berman
author: W. J. Blok
reviewed by: Sheila Oates-Williams
MaRDI profile type: Publication
cites work: Q3775604
cites work: The structure of finite algebras
cites work: Varieties of groups
cites work: SOME INDECOMPOSABLE VARIETIES OF GROUPS
description: scientific article; zbMATH DE number 4055693