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 à Âukasiewicz logic; the letters MV refer to the many-valued logic of à Âukasiewicz. MV-algebras coincide with the class of bounded commutative BCK algebras.
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 (Hájek 1998) as a prelinear commutative bounded integral residuated lattice satisfying the additional identity
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 à Âukasiewicz 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 MV<sub>3</sub> algebra corresponding to the three-valued à Âukasiewicz logic à Â<sub>3</sub>. 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 (both included), that is, the set which is closed under the operations and of the standard MV-algebra; these algebras are usually denoted MV<sub>n</sub>.
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 â G | 0 ⤠x ⤠u }, which becomes an MV-algebra with x â y = min(u, x + y) and ìx = u â x. Furthermore, Chang showed that every linearly ordered MV-algebra is isomorphic to an MV-algebra constructed from a group in this way.
Daniele 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 â G | 0 ⤠x ⤠u } can be equipped with ìx = u â x, x â y = u â§<sub>G</sub> (x + y), and x â y = 0 â¨<sub>G</sub> (x + y â u). This construction establishes a categorical equivalence between lattice-ordered abelian groups with strong unit and MV-algebras.
An effect algebra that is lattice-ordered and has the Riesz decomposition property is an MV-algebra. Conversely, any MV-algebra is a lattice-ordered effect algebra with the Riesz decomposition property.
C. C. Chang devised MV-algebras to study many-valued logics, introduced by Jan à Âukasiewicz in 1920. In particular, MV-algebras form the algebraic semantics of à Âukasiewicz 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 à Âukasiewicz 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 à Âukasiewicz 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 à Âukasiewicz logic in a manner analogous to the way that Boolean algebras characterize classical bivalent logic (see LindenbaumâÂÂTarski algebra).
In 1984, Font, Rodriguez and Torrens introduced the Wajsberg algebra as an alternative model for the infinite-valued à Âukasiewicz logic. Wajsberg algebras and MV-algebras are term-equivalent.<Ref></ref>
In the 1940s, Grigore Moisil introduced his à ÂukasiewiczâÂÂMoisil algebras (LM<sub>n</sub>-algebras) in the hope of giving algebraic semantics for the (finitely) n-valued à Âukasiewicz logic. However, in 1956, Alan Rose discovered that for n âÂÂ¥ 5, the à ÂukasiewiczâÂÂMoisil algebra does not model the à Âukasiewicz n-valued logic. Although C. C. Chang published his MV-algebra in 1958, it is a faithful model only for the âµ<sub>0</sub>-valued (infinitely-many-valued) à ÂukasiewiczâÂÂTarski logic. For the axiomatically more complicated (finitely) n-valued à Âukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MV<sub>n</sub>-algebras. MV<sub>n</sub>-algebras are a subclass of LM<sub>n</sub>-algebras; the inclusion is strict for n âÂÂ¥ 5.
The MV<sub>n</sub>-algebras are MV-algebras that satisfy some additional axioms, just like the n-valued à Âukasiewicz logics have additional axioms added to the âµ<sub>0</sub>-valued logic.
In 1982, Roberto Cignoli published some additional constraints that added to LM<sub>n</sub>-algebras yield proper models for n-valued à Âukasiewicz logic; Cignoli called his discovery proper n-valued à Âukasiewicz algebras. The LM<sub>n</sub>-algebras that are also MV<sub>n</sub>-algebras are precisely Cignoli's proper n-valued à Âukasiewicz 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:
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.