Cardinality bounds for subdirectly irreducible algebras
type: Item
instance of: scholarly article
title: Cardinality bounds for subdirectly irreducible algebras
zbMATH Open document ID: 0871.08007
DOI: 10.1016/0022-4049(95)00137-9
published in: Journal of Pure and Applied Algebra
publication date: 1997-09-29T00:00:00Z
review text: This is a well-written and nicely presented paper that everyone should read who is interested in the trends of modern universal algebra. The author tells the history of the topic, which enlightens the difficulty of the subject and the depth of its proof. Previous attempts to prove that a residually small variety generated by a finite algebra is residually bounded by a finite integer, failed. The reason was undecidability, as was shown by \textit{R. McKenzie} [``The residual bound of a finite algebra is not computable'', Int. J. Algebra Comput. 6, No. 1, 29-48 (1996; Zbl 0844.08010)]. The positive result of this paper was proved by the author in 1991, but remained unpublished, hoping for a complete solution. The result is the following: In a finitely generated residually small variety there is a finite cardinal bound which holds for all subdirectly irreducible algebras, which omits type \textbf{5} and whose monolith is of type \textbf{2}, \textbf{3}, or \textbf{4}. In fact, if the generating algebra has exactly \(n\) elements, then the bound is \(n^{n^{n^{n-2}}}\). As the result shows, the author uses tame congruence theory.
Mathematics Subject Classification ID: 08B26
zbMATH DE Number: 937164
zbMATH Keywords: residually small variety
zbMATH Keywords: subdirectly irreducible algebras
zbMATH Keywords: tame congruence
author: Keith A. Kearnes
reviewed by: Ervin Fried
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00137-9
OpenAlex ID: W2039193053
cites work: Residually Small Varieties with Modular Congruence Lattices
cites work: Q3775604
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cites work: Algebras Whose Congruence Lattices are Distributive.
cites work: Residual bounds for varieties of modules
cites work: AN ORDER-THEORETIC PROPERTY OF THE COMMUTATOR
cites work: Q3751631
cites work: Q4075458
cites work: Residually small varieties
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description: scientific article; zbMATH DE number 937164