MV-algebra

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

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

which satisfies the following identities:

• ${\displaystyle (x\oplus y)\oplus z=x\oplus (y\oplus z),}$
• ${\displaystyle x\oplus 0=x,}$
• ${\displaystyle x\oplus y=y\oplus x,}$
• ${\displaystyle \lnot \lnot x=x,}$
• ${\displaystyle x\oplus \lnot 0=\lnot 0,}$ and
• ${\displaystyle \lnot (\lnot x\oplus y)\oplus y=\lnot (\lnot y\oplus x)\oplus x.}$

By virtue of the first three axioms, ${\displaystyle \langle A,\oplus ,0\rangle }$ 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 ${\displaystyle \langle L,\wedge ,\vee ,\otimes ,\rightarrow ,0,1\rangle }$ satisfying the additional identity ${\displaystyle x\vee y=(x\rightarrow y)\rightarrow y.}$

Examples of MV-algebras[]

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

The two-element MV-algebra is actually the two-element Boolean algebra ${\displaystyle \{0,1\},}$ with ${\displaystyle \oplus }$ coinciding with Boolean disjunction and ${\displaystyle \lnot }$ with Boolean negation. In fact adding the axiom ${\displaystyle x\oplus x=x}$ to the axioms defining an MV-algebra results in an axiomatization of Boolean algebras.

If instead the axiom added is ${\displaystyle x\oplus x\oplus x=x\oplus x}$, 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 ${\displaystyle n}$ equidistant real numbers between 0 and 1 (both included), that is, the set ${\displaystyle \{0,1/(n-1),2/(n-1),\dots ,1\},}$ which is closed under the operations ${\displaystyle \oplus }$ and ${\displaystyle \lnot }$ 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), and x ⊗ y = 0 ∨_G (x + y − 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.^[1]

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 ${\displaystyle \oplus ,\lnot ,}$ and 0) into A. Formulas mapped to 1 (that is, to ${\displaystyle \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 as an alternative model for the infinite-valued Łukasiewicz logic. Wajsberg algebras and MV-algebras are term-equivalent.^[2]

MV_n-algebras[]

This section needs expansion with: the extra axioms. You can help by . (August 2014)

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 a 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 and called MV_n-algebras.^[3] MV_n-algebras are a subclass of LM_n-algebras; the inclusion is strict for n ≥ 5.^[4]

The MV_n-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 published some additional constraints that added to LM_n-algebras yield proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper n-valued Łukasiewicz algebras.^[5] The LM_n-algebras that are also MV_n-algebras are precisely Cignoli’s proper n-valued Łukasiewicz algebras.^[6]

Relation to functional analysis[]

This section needs expansion. You can help by . (November 2012)

MV-algebras were related by 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:

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 an 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: 463–474 doi: 10.1006/jabr.1999.7900.
• 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
• D. Mundici (2011). Advanced Łukasiewicz calculus and MV-algebras. Springer. ISBN 978-94-007-0839-6.
• 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.