Free spectra of nilpotent varieties
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.
free spectrum of a locally finite variety
finitely generated congruence uniform variety
scientific article; zbMATH DE number 4055693