In abstract algebra, a branch of pure mathematics, an MValgebra is an algebraic structure with a binary operation \oplus, a unary operation \neg, and the constant 0, satisfying certain axioms. MValgebras are the algebraic semantics of Łukasiewicz logic; the letters MV refer to the manyvalued logic of Łukasiewicz. MValgebras coincide with the class of bounded commutative BCK algebras.
Definitions
An MValgebra is an algebraic structure \langle A, \oplus, \lnot, 0\rangle, consisting of
which satisfies the following identities:

(x \oplus y) \oplus z = x \oplus (y \oplus z),

x \oplus 0 = x,

x \oplus y = y \oplus x,

\lnot \lnot x = x,

x \oplus \lnot 0 = \lnot 0, and

\lnot ( \lnot x \oplus y)\oplus y = \lnot ( \lnot y \oplus x) \oplus x.
By virtue of the first three axioms, \langle A, \oplus, 0 \rangle is a commutative monoid. Being defined by identities, MValgebras form a variety of algebras. The variety of MValgebras is a subvariety of the variety of BLalgebras and contains all Boolean algebras.
An MValgebra can equivalently be defined (Hájek 1998) as a prelinear commutative bounded integral residuated lattice \langle L, \wedge, \vee, \otimes, \rightarrow, 0, 1 \rangle satisfying the additional identity x \vee y = (x \rightarrow y) \rightarrow y.
Examples of MValgebras
A simple numerical example is A=[0,1], with operations x \oplus y = \min(x+y,1) and \lnot x=1x. In mathematical fuzzy logic, this MValgebra is called the standard MValgebra, as it forms the standard realvalued semantics of Łukasiewicz logic.
The trivial MValgebra has the only element 0 and the operations defined in the only possible way, 0\oplus0=0 and \lnot0=0.
The twoelement MValgebra is actually the twoelement Boolean algebra \{0,1\}, with \oplus coinciding with Boolean disjunction and \lnot with Boolean negation. In fact adding the axiom x \oplus x = x to the axioms defining an MValgebra results in an axiomantization of Boolean algebras.
If instead the axiom added is x \oplus x \oplus x = x \oplus x, then the axioms define the MV_{3} algebra corresponding to the threevalued Łukasiewicz logic Ł_{3}. Other finite linearly ordered MValgebras are obtained by restricting the universe and operations of the standard MValgebra to the set of n equidistant real numbers between 0 and 1 (both included), that is, the set \{0,1/(n1),2/(n1),\dots,1\}, which is closed under the operations \oplus and \lnot of the standard MValgebra; these algebras are usually denoted MV_{n}.
Another important example is Chang's MValgebra, consisting just of infinitesimals (with the order type ω) and their coinfinitesimals.
Chang also constructed an MValgebra 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 MValgebra with x ⊕ y = min(u, x+y) and ¬x = u−x. Furthermore, Chang showed that every linearly ordered MValgebra is isomorphic to an MValgebra constructed from a group in this way.
D. Mundici extended the above construction to abelian latticeordered 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 latticeordered abelian groups with strong unit and MValgebras.
Relation to Łukasiewicz logic
C. C. Chang devised MValgebras to study manyvalued logics, introduced by Jan Łukasiewicz in 1920. In particular, MValgebras form the algebraic semantics of Łukasiewicz logic, as described below.
Given an MValgebra A, an Avaluation is a homomorphism from the algebra of propositional formulas (in the language consisting of \oplus,\lnot, and 0) into A. Formulas mapped to 1 (or \lnot0) for all Avaluations are called Atautologies. If the standard MValgebra over [0,1] is employed, the set of all [0,1]tautologies determines socalled infinitevalued Łukasiewicz logic.
Chang's (1958, 1959) completeness theorem states that any MValgebra equation holding in the standard MValgebra over the interval [0,1] will hold in every MValgebra. Algebraically, this means that the standard MValgebra generates the variety of all MValgebras. Equivalently, Chang's completeness theorem says that MValgebras characterize infinitevalued Łukasiewicz logic, defined as the set of [0,1]tautologies.
The way the [0,1] MValgebra characterizes all possible MValgebras parallels the wellknown fact that identities holding in the twoelement Boolean algebra hold in all possible Boolean algebras. Moreover, MValgebras characterize infinitevalued Łukasiewicz logic in a manner analogous to the way that Boolean algebras characterize classical bivalent logic (see LindenbaumTarski algebra).
In 1984, Font, Rodriguez and Torrens introduced the Wajsberg algebra as an alternative model for the infinitevalued Łukasiewicz logic. Wajsberg algebras and MValgebras are isomorphic.^{[1]}
MV_{n}algebras
In the 1940s Grigore Moisil introduced his Łukasiewicz–Moisil algebras (LM_{n}algebras) in the hope of giving algebraic semantics for the (finitely) nvalued Łukasiewicz logic. However, in 1956 Alan Rose discovered that for n ≥ 5, the Łukasiewicz–Moisil algebra does not model the Łukasiewicz nvalued logic. Although C. C. Chang published his MValgebra in 1958, it is faithful model only for the ℵ_{0}valued (infinitelymanyvalued) Łukasiewicz–Tarski logic. For the axiomatically more complicated (finitely) nvalued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MV_{n}algebras.^{[2]} MV_{n}algebras are a subclass of LM_{n}algebras; the inclusion is strict for n ≥ 5.^{[3]}
The MV_{n}algebras are MValgebras which satisfy some additional axioms, just like the nvalued Łukasiewicz logics have additional axioms added to the ℵ_{0}valued logic.
In 1982 Roberto Cignoli published some additional constraints that added to LM_{n}algebras are proper models for nvalued Łukasiewicz logic; Cignoli called his discovery proper nvalued Łukasiewicz algebras.^{[4]} The LM_{n}algebras that are also MV_{n}algebras are precisely Cignoli’s proper nvalued Łukasiewicz algebras.^{[5]}
Relation to functional analysis
MValgebras were related by Daniele Mundici to approximately finitedimensional C*algebras by establishing a bijective correspondence between all isomorphism classes of AF C*algebras with latticeordered dimension group and all isomorphism classes of countable MV algebras. Some instances of this correspondence include:
Countable MV algebra

AF C*algebra

{0, 1}

ℂ

{0, 1/n, ..., 1 }

M_{n}(ℂ), i.e. n×n complex matrices

finite

finitedimensional

boolean

commutative

In software
There are multiple frameworks implementing fuzzy logic (type II), and most of them implement what has been called a multiadjoint logic. This is no more than the implementation of a MValgebra.
References

^ http://journal.univagora.ro/download/pdf/28.pdf citing J. M. Font, A. J. Rodriguez, A. Torrens, Wajsberg Algebras, Stochastica, VIII, 1, 531, 1984

^ Lavinia Corina Ciungu (2013). Noncommutative MultipleValued Logic Algebras. Springer. pp. vii–viii.

^ Iorgulescu, A.: Connections between MV_{n}algebras and nvalued Łukasiewicz–Moisil algebras—I. Discrete Math. 181, 155–177 (1998) doi:10.1016/S0012365X(97)000526

^ R. Cignoli, Proper nValued Łukasiewicz Algebras as SAlgebras of Łukasiewicz nValued Propositional Calculi, Studia Logica, 41, 1982, 316, doi:10.1007/BF00373490

^ http://journal.univagora.ro/download/pdf/28.pdf

Chang, C. C. (1958) "Algebraic analysis of manyvalued logics," Transactions of the American Mathematical Society 88: 476–490.

 (1959) "A new proof of the completeness of the Lukasiewicz axioms," Transactions of the American Mathematical Society 88: 74–80.

Cignoli, R. L. O., D'Ottaviano, I. M. L., Mundici, D. (2000) Algebraic Foundations of Manyvalued Reasoning. Kluwer.

Di Nola A., Lettieri A. (1993) "Equational characterization of all varieties of MValgebras," Journal of Algebra 221: 123–131.

Hájek, Petr (1998) Metamathematics of Fuzzy Logic. Kluwer.

Mundici, D.: Interpretation of AF C*algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986) doi:10.1016/00221236(86)900157
Further reading
External links
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