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. 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 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 $\oplus,\lnot,$ 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.

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) 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:

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

In software

Further information: Multi-adjoint logic programming

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

• Chang, C. C. (1958) "Algebraic analysis of many-valued 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 Many-valued Reasoning. Kluwer.
• Di Nola A., Lettieri A. (1993) "Equational characterization of all varieties of MV-algebras," 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:10.1016/0022-1236(86)90015-7

Further reading

• Daniele Mundici, MV-ALGEBRAS. A short tutorial
• Mundici, D. The C*-Algebras of Three-Valued Logic. Logic Colloquium ’88, Proceedings of the Colloquium held in Padova 61–77 (1989). doi:10.1016/s0049-237x(08)70262-3
• Cabrer, L. M. & Mundici, D. A Stone-Weierstrass theorem for MV-algebras and unital ℓ-groups. Journal of Logic and Computation (2014). doi:10.1093/logcom/exu023
• Olivia Caramello, Anna Carla Russo (2014) The Morita-equivalence between MV-algebras and abelian ℓ-groups with strong unit

External links

• Stanford Encyclopedia of Philosophy: "Many-valued logic"—by Siegfried Gottwald.