Congruence lattices of powers of an algebra
type: Item
instance of: scholarly article
title: Congruence lattices of powers of an algebra
zbMATH Open document ID: 0686.08008
DOI: 10.1007/BF01211839
author: Ross Willard
published in: Algebra Universalis
publication date: 1989-01-01T00:00:00Z
review text: Let \(\mathcal P\) be a property which can be attributed to 0-1 sublattices of the equivalence lattice on an arbitrary set. It was already proved by S. Burris and the author that for every integer \(k\geq 2\) there is an \(n\geq 1\) such that for every algebra \(A\) of cardinality \(k\), if \(\text{Con}(A^ m)\) satisfies \(\mathcal P\) for all \(m\leq n\) then \(\text{Con}(A^ m)\) satisfies \(P\) for all \(m\). Let \(n_{\mathcal P}(k)\) denote the least such \(n\). The paper evaluates the function \(n_{\mathcal P}\) for \(p=\) permutability, distributivity, arithmeticity, weak distributivity, modularity, and the property of being skew-free (= the Fraser-Horn property). Main results: (i) \(n_{\text{Perm}}(k)\leq k^ 3\); (ii) \(n_{\text{Dist}}(k)\leq k^{k+1}\); (iii) \(n_{\text{Skew-free}}(k)\leq k^3 + k^2 - k\); (iv) \(n_{\text{Mod}}(k)\leq k^{4m(k)}\), where \(m(k)=k^{(k^ 4-k^3 + k^2)} - 1\).
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 4123795
zbMATH Keywords: power of algebra
zbMATH Keywords: congruence lattice
zbMATH Keywords: permutability
zbMATH Keywords: distributivity
zbMATH Keywords: arithmeticity
zbMATH Keywords: weak distributivity
zbMATH Keywords: modularity
zbMATH Keywords: skew-free
zbMATH Keywords: Fraser-Horn property
reviewed by: Jaromír Duda
MaRDI profile type: Publication
cites work: Remarks on the Fraser-Horn property
cites work: Q3934450
cites work: Finitely Many Primitive Positive Clones
cites work: A Characterization of Modularity for Congruence Lattices of Algebras*
cites work: Algebras Whose Congruence Lattices are Distributive.
cites work: Distributivity and Permutability of Congruence Relations in Equational Classes of Algebras
cites work: Local Malcev Conditions
full work available at URL: https://doi.org/10.1007/bf01211839
OpenAlex ID: W2113485857
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description: scientific article; zbMATH DE number 4123795