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.

Definitions

An MV-algebra is an algebraic structure consisting of
which satisfies the following identities:
By virtue of the first three axioms, 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 as a prelinear commutative bounded integral residuated lattice satisfying the additional identity

Examples of MV-algebras

A simple numerical example is with operations and 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, and
The two-element MV-algebra is actually the two-element Boolean algebra with coinciding with Boolean disjunction and with Boolean negation. In fact adding the axiom to the axioms defining an MV-algebra results in an axiomatization of Boolean algebras.
If instead the axiom added is, 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 equidistant real numbers between 0 and 1, that is, the set which is closed under the operations and of the standard MV-algebra; these algebras are usually denoted MVn.
Another important example is Chang's MV-algebra, consisting just of infinitesimals 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 as, which becomes an MV-algebra with x ⊕ y = min 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 unit u, then the "unit interval" can be equipped with ¬x = u − x, x ⊕ y = u ∧G, and x ⊗ y = 0 ∨G. This construction establishes a categorical equivalence between lattice-ordered abelian groups with strong unit and MV-algebras.

Relation to Łukasiewicz logic

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 into A. Formulas mapped to 1 for all A-valuations are called A-tautologies. If the standard MV-algebra over is employed, the set of all -tautologies determines so-called infinite-valued Łukasiewicz logic.
Chang's completeness theorem states that any MV-algebra equation holding in the standard MV-algebra over the interval 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 -tautologies.
The way the 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.
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.

MV''n''-algebras

In the 1940s Grigore Moisil introduced his Łukasiewicz–Moisil algebras in the hope of giving algebraic semantics for the 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 Łukasiewicz–Tarski logic. For the axiomatically more complicated 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.

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 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 algebraapproximately finite-dimensional C*-algebra
ℂ
Mn, i.e. n×n complex matrices
finitefinite-dimensional
booleancommutative

In software

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