# MV-algebra

In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation $\oplus$, a unary operation $\neg$, and the constant $0$, 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 $\langle A, \oplus, \lnot, 0\rangle,$ consisting of

• a non-empty set $A,$
• a binary operation $\oplus$ on $A,$
• a unary operation $\lnot$ on $A,$ and
• a constant $0$ denoting a fixed element of $A,$

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, 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 $\langle L, \wedge, \vee, \otimes, \rightarrow, 0, 1 \rangle$ satisfying the additional identity $x \vee y = (x \rightarrow y) \rightarrow y.$

## Examples of MV-algebras

A simple numerical example is $A=[0,1],$ with operations $x \oplus y = \min(x+y,1)$ and $\lnot x=1-x.$ 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, $0\oplus0=0$ and $\lnot0=0.$

The two-element MV-algebra is actually the two-element 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 MV-algebra 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 MV3 algebra corresponding to the three-valued Łukasiewicz logic Ł3[citation needed]. 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 $\{0,1/(n-1),2/(n-1),\dots,1\},$ which is closed under the operations $\oplus$ and $\lnot$ of the standard MV-algebra; these algebras are usually denoted MVn.

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 { xG | 0 ≤ xu }, which becomes an MV-algebra with xy = min(u, x+y) and ¬x = ux. 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" { xG | 0 ≤ xu } can be equipped with ¬x = ux, xy = uG (x+y), xy = 0∨G(x+yu). 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 $\oplus,\lnot,$ and 0) into A. Formulas mapped to 1 (or $\lnot$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.[1]

### MVn-algebras

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 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.[2] MVn-algebras are a subclass of LMn-algebras; the inclusion is strict for n ≥ 5.[3]

The MVn-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 LMn-algebras are proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper n-valued Łukasiewicz algebras.[4] The LMn-algebras that are also MVn-algebras are precisely Cignoli’s proper n-valued Łukasiewicz algebras.[5]

## 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:

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

## 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.

## References

1. ^ http://journal.univagora.ro/download/pdf/28.pdf citing J. M. Font, A. J. Rodriguez, A. Torrens, Wajsberg Algebras, Stochastica, VIII, 1, 5-31, 1984
2. ^ Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. pp. vii–viii. ISBN 978-3-319-01589-7.
3. ^ Iorgulescu, A.: Connections between MVn-algebras and n-valued Łukasiewicz–Moisil algebras—I. Discrete Math. 181, 155–177 (1998) doi:10.1016/S0012-365X(97)00052-6
4. ^ R. Cignoli, Proper n-Valued Łukasiewicz Algebras as S-Algebras of Łukasiewicz n-Valued Propositional Calculi, Studia Logica, 41, 1982, 3-16, doi:10.1007/BF00373490