MV-algebra

In abstract algebra, a branch of pure mathematics, an MV algebra is an algebraic structure with a binary operation , a unary operation , and the constant , satisfying certain axioms. MV algebras are the algebraic semantics of Łukasiewicz logic; the letters MV refer to the many valued logic of Łukasiewicz. MV algebras coincide with the class of bounded commutative BCK algebras.

C. C. Chang devised MV algebras to study many valued logics, introduced by Jan Łukasiewicz in 1920. In particular, MV algebras form the algebraic semantics of Łukasiewicz logic, as described below. Given an MV algebra A, an A valuation is a homomorphism from the algebra of propositional formulas (in the language consisting of and 0) into A. Formulas mapped to 1 (that is, to 0) for all A valuations are called A tautologies. If the standard MV algebra over [0,1] is employed, the set of all [0,1] tautologies determines so called infinite valued Łukasiewicz logic. Chang's (1958, 1959) completeness theorem states that any MV algebra equation holding in the standard MV algebra over the interval [0,1] will hold in every MV algebra. Algebraically, this means that the standard MV algebra generates the variety of all MV algebras. Equivalently, Chang's completeness theorem says that MV algebras characterize infinite valued Łukasiewicz logic, defined as the set of [0,1] tautologies. The way the [0,1] MV algebra characterizes all possible MV algebras parallels the well known fact that identities holding in the two element Boolean algebra hold in all possible Boolean algebras. Moreover, MV algebras characterize infinite valued Łukasiewicz logic in a manner analogous to the way that Boolean algebras characterize classical bivalent logic (see Lindenbaum–Tarski algebra). In 1984, Font, Rodriguez and Torrens introduced the Wajsberg algebra as an alternative model for the infinite valued Łukasiewicz logic. Wajsberg algebras and MV algebras are term equivalent.

In the 1940s Grigore Moisil introduced his Łukasiewicz–Moisil algebras (LMn algebras) in the hope of giving algebraic semantics for the (finitely) n valued Łukasiewicz logic. However, in 1956 Alan Rose discovered that for n ≥ 5, the Łukasiewicz–Moisil algebra does not model the Łukasiewicz n valued logic. Although C. C. Chang published his MV algebra in 1958, it is a faithful model only for the ℵ0 valued (infinitely many valued) Łukasiewicz–Tarski logic. For the axiomatically more complicated (finitely) n valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MVn algebras. MVn algebras are a subclass of LMn algebras; the inclusion is strict for n ≥ 5. The MVn algebras are MV algebras that satisfy some additional axioms, just like the n valued Łukasiewicz logics have additional axioms added to the ℵ0 valued logic. In 1982 Roberto Cignoli published some additional constraints that added to LMn algebras yield proper models for n valued Łukasiewicz logic; Cignoli called his discovery proper n valued Łukasiewicz algebras. The LMn algebras that are also MVn algebras are precisely Cignoli’s proper n valued Łukasiewicz algebras.

MV algebras were related by Daniele Mundici to approximately finite dimensional C* algebras by establishing a bijective correspondence between all isomorphism classes of approximately finite dimensional C* algebras with lattice ordered dimension group and all isomorphism classes of countable MV algebras. Some instances of this correspondence include:

Countable MV algebra approximately finite dimensional C* algebra {0, 1} ℂ {0, 1/n, , 1 } Mn(ℂ), i. e. n×n complex matrices finite finite dimensional boolean commutative

There are multiple frameworks implementing fuzzy logic (type II), and most of them implement what has been called a multi adjoint logic. This is no more than the implementation of an MV algebra.