MV algebra

Definitions

An MV-algebra is an algebraic structure ⟨ A , ⊕ , ¬ , 0 ⟩ , consisting of

which satisfies the following identities:

By virtue of the first three axioms, ⟨ A , ⊕ , 0 ⟩ is a commutative monoid. Being defined by identities, MV-algebras form a variety of algebras. The variety of MV-algebras is a subvariety of the variety of BL-algebras and contains all Boolean algebras.

An MV-algebra can equivalently be defined (Hájek 1998) as a prelinear commutative bounded integral residuated lattice ⟨ L , ∧ , ∨ , ⊗ , → , 0 , 1 ⟩ satisfying the additional identity x ∨ y = ( x → y ) → y .

Examples of MV-algebras

A simple numerical example is A = [ 0 , 1 ] , with operations x ⊕ y = min ( x + y , 1 ) and ¬ x = 1 − x . In mathematical fuzzy logic, this MV-algebra is called the standard MV-algebra, as it forms the standard real-valued semantics of Łukasiewicz logic.

The trivial MV-algebra has the only element 0 and the operations defined in the only possible way, 0 ⊕ 0 = 0 and ¬ 0 = 0.

The two-element MV-algebra is actually the two-element Boolean algebra { 0 , 1 } , with ⊕ coinciding with Boolean disjunction and ¬ with Boolean negation. In fact adding the axiom x ⊕ x = x to the axioms defining an MV-algebra results in an axiomatization of Boolean algebras.

If instead the axiom added is x ⊕ x ⊕ x = x ⊕ x , then the axioms define the MV₃ algebra corresponding to the three-valued Łukasiewicz logic Ł₃. Other finite linearly ordered MV-algebras are obtained by restricting the universe and operations of the standard MV-algebra to the set of n equidistant real numbers between 0 and 1 (both included), that is, the set { 0 , 1 / ( n − 1 ) , 2 / ( n − 1 ) , … , 1 } , which is closed under the operations ⊕ and ¬ of the standard MV-algebra; these algebras are usually denoted MV_n.

Another important example is Chang's MV-algebra, consisting just of infinitesimals (with the order type ω) and their co-infinitesimals.

Chang also constructed an MV-algebra from an arbitrary totally ordered abelian group G by fixing a positive element u and defining the segment [0, u] as { x ∈ G | 0 ≤ x ≤ u }, which becomes an MV-algebra with x ⊕ y = min(u, x+y) and ¬x = u−x. Furthermore, Chang showed that every linearly ordered MV-algebra is isomorphic to an MV-algebra constructed from a group in this way.

D. Mundici extended the above construction to abelian lattice-ordered groups. If G is such a group with strong (order) unit u, then the "unit interval" { x ∈ G | 0 ≤ x ≤ u } can be equipped with ¬x = u−x, x ⊕ y = u∧_G (x+y), x ⊗ y = 0∨_G(x+y−u). This construction establishes a categorical equivalence between lattice-ordered abelian groups with strong unit and MV-algebras.

Relation to Łukasiewicz logic

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 (or ¬ 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 isomorphic.

MVn-algebras

In the 1940s Grigore Moisil introduced his Łukasiewicz–Moisil algebras (LM_n-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 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 MV_n-algebras. MV_n-algebras are a subclass of LM_n-algebras; the inclusion is strict for n ≥ 5.

The MV_n-algebras are MV-algebras which 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 LM_n-algebras are proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper n-valued Łukasiewicz algebras. The LM_n-algebras that are also MV_n-algebras are precisely Cignoli’s proper n-valued Łukasiewicz algebras.

Relation to functional analysis

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

In software

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 a MV-algebra.