Congruence join semidistributivity is equivalent to a congruence identity.
type: Item
instance of: scholarly article
title: Congruence join semidistributivity is equivalent to a congruence identity.
zbMATH Open document ID: 1063.08008
DOI: 10.1007/PL00000351
published in: Algebra Universalis
publication date: 2005-04-19T00:00:00Z
review text: A lattice is join semidistributive if it satisfies the implication \[ \alpha \vee \beta = \alpha \vee \gamma \Rightarrow \alpha \vee \beta = \alpha \vee (\beta \wedge \gamma ). \] For \(\alpha ,\beta ,\gamma \in \text{Con}{\mathcal A}\) define recursively \(\beta ^0 =\beta , \, \gamma ^0 =\gamma , \,\beta ^{n+1} =\beta \wedge (\alpha \vee \gamma ^n), \, \gamma ^{n+1} = \gamma \wedge ( \alpha \vee \beta ^n)\). The main result: Let \({\mathcal V}\) be a locally finite variety. Then \({\mathcal V}\) is congruence join semidistributive if for some integer \(n\), \({\mathcal V}\) satisfies \[ \alpha \vee (\beta \wedge \gamma ) = (\alpha \vee \beta ^n) \wedge (\alpha \vee \gamma ^n) \] as a congruence identity. This identity is strong enough to force join semidistributivity in any lattice.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 08B10
zbMATH DE Number: 2158750
zbMATH Keywords: congruence identity
zbMATH Keywords: congruence variety
zbMATH Keywords: semidistributivity
author: Keith A. Kearnes
MaRDI profile type: Publication
full work available at URL: pl00000351
OpenAlex ID: W2022941646
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description: scientific article; zbMATH DE number 2158750