MV-algebras: a variety for magnitudes with archimedean units

Abstract

Chang's MV-algebras, on the one hand, are the algebras of the infinite-valued Lukasiewicz calculus and, on the other hand, are categorically equivalent to abelian latticeordered groups with a distinguished strong unit, for short, unital -groups. The latter are a modern mathematization of the time-honored euclidean magnitudes with an archimedean unit. While for magnitudes the unit is no less important than the zero element, its archimedean property is not even definable in first-order logic. This gives added interest to the equivalent representation of unital -groups via the equational class of MV-algebras. In this paper we survey several applications of this equivalence, and various properties of the variety of MV-algebras. PrologueIn the last twenty years the number of papers devoted to Chang's MV-algebras [14] has been increasing so rapidly that, since the year 2000 the AMS Classification Index contains the special item 06D35 for MV-algebras. To quote just a handful of books, the monograph [25] by Cignoli et al., is entirely devoted to MValgebras, Hajek's monograph [56] and Gottwald's book [54] devote ample space to these algebras. As shown in [37] and [94], MV-algebras also provide an interesting example of "quantum structures". The Handbook of Measure Theory [89] includes several chapters on MV-algebraic measure theory. As the Lindenbaum algebras of Lukasiewicz infinite-valued logic, MV-algebras are also considered in many surveys, e.g., [21,68,87]. In this paper we will present several main topics in the theory of MV-algebras which are of greater potential interest for the universal algebraist. We apologize to authors and readers for all omissions, mainly due to lack of space.

A characterization of free generating sets in MV-algebras

“…We assume familiarity with MV-algebras, for which we refer to [4] and [5]. As noted in [4, 1.7], the current definition of MV-algebra is due to Mangani [7], who greatly simplified Chang's original definition [1].…”

Extending Łukasiewicz Logics with a Modality: Algebraic Approach to Relational Semantics

“…This somehow obvious result can be seen as a step towards the construction of the canonical model and possible completeness theorems for manyvalued modal logics and many-valued Kripke models. We refer the reader to [8] or [18] for an introduction to the variety of MV-algebras.…”

A general Glivenko theorem

“…Recall [14] that an MV-algebra can be characterized as a commutative monoid X with an involution x → x * such that 0 := 1 * satisfies x0 = 0, and…”