Komori's characterization and top varieties of GMV-algebras (Q1013984)

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.