On finite bases for varieties and pseudovarieties
type: Item
instance of: scholarly article
title: On finite bases for varieties and pseudovarieties
zbMATH Open document ID: 0655.08006
DOI: 10.1007/BF01229976
author: Robert Cacioppo
published in: Algebra Universalis
publication date: 1988-01-01T00:00:00Z
review text: Let P be a partition of an algebra A, and let S be a set of distinct representatives for P. Then the (S,P)-reduction of A is the algebra on S of the same type as A with operations defined via representatives modulo P. Such a reduction may or may not preserve composite terms; if it does preserve all terms of length at most n, it is called an n-reduction. The author proves that a variety of finite type is finitely based if and only if it is closed under n-reductions for some finite n. By related means, he shows that certain classes of finite algebras generating locally finite varieties generate nonfinitely based pseudovarieties; as examples, he mentions \textit{P. Perkins}' 6-element semigroup [J. Algebra 11, 298-314 (1969; Zbl 0186.03401)] and \textit{R. C. Lyndon}'s 7-element groupoid [Proc. Am. Math. Soc. 5, 8-9 (1954; Zbl 0055.027)].
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 4068958
zbMATH Keywords: equational basis
zbMATH Keywords: finitely based varieties
zbMATH Keywords: n-reduction
zbMATH Keywords: pseudovarieties
reviewed by: Manfred Armbrust
MaRDI profile type: Publication
cites work: Equational classes of modular lattices
cites work: Q3934450
cites work: On pseudovarieties
cites work: Identities in Finite Algebras
cites work: Q3658046
cites work: Bases for equational theories of semi-groups
cites work: The Birkhoff theorem for finite algebras
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description: scientific article; zbMATH DE number 4068958