Komori's characterization and top varieties of GMV-algebras
type: Item
instance of: scholarly article
title: Komori's characterization and top varieties of GMV-algebras
zbMATH Open document ID: 1168.06007
author: Anatolij Dvurečenskij
author: W. Charles Holland
published in: Algebra Universalis
publication date: 2009-04-24T00:00:00Z
review text: Chang invented MV-algebras to give an algebraic proof of the completeness of Łukasiewicz infinite-valued propositional logic, GMV-algebras were introduced by Georgescu and Iorgulescu and, independently, by Rachůnek as a noncommutative generalization of MV-algebras. In his paper [J. Funct. Anal. 65, 15--63 (1986; Zbl 0597.46059)], the present reviewer proved that MV-algebras are categorically equivalent to unital lattice-ordered abelian groups. This result was extended by \textit{A. Dvurečenskij} [J. Aust. Math. Soc. 72, No. 3, 427--445 (2002; Zbl 1027.06014)]. He proved that unital lattice-ordered groups are categorically equivalent to GMV-algebras. One can now naturally speak of ``varieties'' of unital \(l\)-groups via their GMV-algebraic counterparts. The classification of varieties of MV-algebras was achieved by Komori (see, e.g., the monograph [\textit{R. L. O. Cignoli, I. M. L. D'Ottaviano} and \textit{D. Mundici}, Algebraic foundations of many-valued reasoning. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)]). The present paper is a generalization of Komori's classification: within the variety of GMV-algebras such that every maximal ideal is normal, the authors characterize the proper top varieties and provide equational bases for these varieties. They also show that there are only countably many such top varieties, each of them having uncountably many subvarieties. Using coproducts, the amalgamation property is shown to fail for those GMV-algebras that can be split into \(n+1\) comparable slices.
reviewed by: Daniele Mundici
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 03B50
zbMATH DE Number: 5547245
zbMATH Keywords: MV-algebra
zbMATH Keywords: GMV-algebra
zbMATH Keywords: unital \(\ell \)-group
zbMATH Keywords: representation
zbMATH Keywords: categorical equivalence
zbMATH Keywords: top variety
zbMATH Keywords: state
zbMATH Keywords: Komori's characterization
MaRDI profile type: Publication
full work available at URL: s00012-008-2085-x
OpenAlex ID: W2073699586
DOI: 10.1007/S00012-008-2085-X
description: scientific article; zbMATH DE number 5547245