Varieties generated by unital abelian \(\ell\)-groups
type: Item
instance of: scholarly article
title: Varieties generated by unital abelian \(\ell\)-groups
zbMATH Open document ID: 1300.06003
published in: Journal of Pure and Applied Algebra
publication date: 2014-08-26T00:00:00Z
reviewed by: Daniele Mundici
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 6334331
zbMATH Keywords: MV-algebra
zbMATH Keywords: equational class
zbMATH Keywords: abelian lattice-ordered group
zbMATH Keywords: unit
MaRDI profile type: Publication
full work available at URL: j.jpaa.2014.04.015
OpenAlex ID: W1991541491
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DOI: 10.1016/J.JPAA.2014.04.015
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author: William Young
review text: This paper deals with the relationships between lattice-ordered abelian groups (\(l\)-groups for short) and MV-algebras. While Chang's MV-algebras are the algebraic semantics for Łukasiewicz infinite-valued propositional logic, \(l\)-groups provide a modern equational framework for the classical theory of magnitudes. In the theory of unital \(l\)-groups one also takes into account the unit of measurent; as a generalization, positively pointed abelian \(l\)-groups do not require the unit to be an Archimedean element.NEWLINENEWLINE As proved by the present reviewer in his paper [J. Funct. Anal. 65, 15--63 (1986; Zbl 0597.46059)], MV-algebras are categorically equivalent to unital \(l\)-groups. The Archimedean property of the unit in unital \(l\)-groups make their class first-order undefinable.NEWLINENEWLINE On the other hand, the author proves that the subvariety lattice of the variety of positively pointed abelian \(l\)-groups (with the exception of the trivial variety) is isomorphic to the subvariety lattice of MV-algebras. Further (with the exception of the unique atom), each subvariety of positively pointed abelian \(l\)-groups is generated by its unital members.
description: scientific article; zbMATH DE number 6334331