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

which satisfies the following identities:

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 axiomantization 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{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[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 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 ${\displaystyle \oplus ,\lnot ,}$ and 0) into A. Formulas mapped to 1 (or ${\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 Wajsberg algebra as an alternative model for the infinite-valued Łukasiewicz logic. Wajsberg algebras and MV-algebras are isomorphic.[1]

MVn-algebras

Template:Expand section 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

Template:Expand section 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

Template:Rellink 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. {{#invoke:citation/CS1|citation |CitationClass=book }}
3. Iorgulescu, A.: Connections between MVn-algebras and n-valued Łukasiewicz–Moisil algebras—I. Discrete Math. 181, 155–177 (1998) Template:Hide in printTemplate:Only in print
4. R. Cignoli, Proper n-Valued Łukasiewicz Algebras as S-Algebras of Łukasiewicz n-Valued Propositional Calculi, Studia Logica, 41, 1982, 3-16, Template:Hide in printTemplate:Only in print