Amalgamation in small varieties of lattices
type: Item
instance of: scholarly article
title: Amalgamation in small varieties of lattices
zbMATH Open document ID: 0733.06003
DOI: 10.1016/0022-4049(90)90143-6
author: Bjarni Jónsson
published in: Journal of Pure and Applied Algebra
publication date: 1990-01-01T00:00:00Z
review text: The amalgamation class Amal(\({\mathcal V})\) of the variety \({\mathcal V}\) was introduced by H. Lakser and the reviewer as the class of all algebras in \({\mathcal V}\) over which one can amalgamate in \({\mathcal V}\); thus, Amal(\({\mathcal V})\) measures to what extent the Amalgamation Property holds (fails) in \({\mathcal V}.\) The author's abstract: ``A property used by G. Grätzer and H. Lakser to describe the amalgamation class of a finitely generated variety of pseudo-complemented distributive lattices, and a property used by C. Bergman in his investigations of the amalgamation classes of varieties that are finitely generated, congruence distributive, and semi-simple, are here applied to arbitrary finitely generated, congruence distributive varieties. The Grätzer-Lakser property is shown to characterize the amalgamation class of a finitely generated class \({\mathcal V}\) of lattices. For finite lattices, this gives an effective test for membership in the amalgamation class of \({\mathcal V}\). For \({\mathcal V}={\mathcal N}\), the variety generated by the pentagon, an even simpler test is found.'' The last sentence of the author's abstract refers to the following elegant result: A finite lattice \(A\in {\mathcal N}\) belongs to Amal(\({\mathcal N})\) iff A is a subdirect power of N and A does not have the three element chain as a homomorphic image.
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 08B10
zbMATH DE Number: 4212117
zbMATH Keywords: finitely generated varieties of lattices
zbMATH Keywords: amalgamation class
zbMATH Keywords: Amalgamation Property
zbMATH Keywords: congruence distributive varieties
zbMATH Keywords: pentagon
MaRDI profile type: Publication
full work available at URL: 0022-4049(90)90143-6
OpenAlex ID: W2040956074
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description: scientific article; zbMATH DE number 4212117