zbMATH — the first resource for mathematics

Free lattice-ordered abelian groups and varieties of MV-algebras. (English) Zbl 0827.06012
Proceedings of the IX Latin American symposium on mathematical logic, Bahía Blanca, Argentina, August 3-8, 1992. Part 1. Bahía Blanca: Universidad Nacional del Sur, Notas Logica Mat. 38, 113-118 (1993).
C. C. Chang introduced MV-algebras in 1958 as the Lindenbaum algebras of the infinite-valued sentential calculus of Lukasiewicz. Following the reviewer’s paper “Interpretation of AF \(C^*\)-algebras in Lukasiewicz sentential calculus” [J. Funct. Anal. 65, 15-63 (1986; Zbl 0597.46059)], a structure \(A= (A,0, 1, {}^*, \oplus)\) is called an MV-algebra iff \((A, 0, \oplus)\) is an abelian monoid, and \(x\oplus 1=1\), \(0^*=1\), \(1^*=0\), \((x^* \oplus y)^* \oplus y= (y^* \oplus x)^* \oplus x\); in the same paper it is shown that MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit. Chang proved that the variety of MV-algebras is generated by the rational unit interval \([0,1]\) equipped with negation \(1-x\) and truncated addition: he used quantifier elimination for totally ordered divisible abelian groups. Using logic-syntactic machinery, Rose and Rosser had previously proved an equivalent version of this theorem. Very recently, G. Panti [J. Symb. Log. 60, No. 2, 563-578 (1995)] gave another proof using the De Concini Procesi theorem on elimination of points of indeterminacy in toric varieties. The author gives a simpler proof of the theorem using the above-mentioned categorical equivalence, together with Weinberg’s well-known representation of free abelian lattice-groups as subdirect products of copies of the additive group \(\mathbb{Z}\) of integers with natural ordering. (The first elementary proof of the theorem is forthcoming, jointly by Cignoli and the present reviewer, in a special issue of Studia Logica.) Using the completeness of the theory of certain classes of totally ordered abelian groups, Komori characterized all subvarieties of MV-algebras. The author proves the same result, in a much simpler way, again making use of Weinberg’s representation theorem.
For the entire collection see [Zbl 0812.00017].
Reviewer: D.Mundici (Milano)

06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
03B50 Many-valued logic