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 $eg$, and the constant $0$, satisfying certain axioms. MV-algebras are models of Łukasiewicz logic; the letters MV refer to "multi-valued" logic of Łukasiewicz.

Definitions

An MV-algebra is an algebraic structure $langle\; A,\; oplus,\; lnot,\; 0\; angle,$ 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, $langle\; A,\; oplus,\; 0\; angle$ 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 $langle\; L,\; wedge,\; vee,\; otimes,\; ightarrow,\; 0,\; 1\; angle$ satisfying the additional identity $x\; vee\; y\; =\; (x\; ightarrow\; y)\; ightarrow\; y.$

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, $0oplus0=0$ 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.

Other finite linearly ordered MV-algebras are obtained by restricting the universe and operations of the standard MV-algebra to the set of $n+1$ equidistant real numbers between 0 and 1 (both included), that is, the set $\{0,1/n,2/n,dots,1\},$ which is closed under the operations $oplus$ and $lnot$ of the standard MV-algebra.

Another important example is "Chang's MV-algebra", consisting just of infinitesimals (with the order type ω) and their co-infinitesimals.

Relation to Łukasiewicz logic

Chang devised MV-algebras to study multi-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).

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.

External links

* Stanford Encyclopedia of Philosophy: " [http://plato.stanford.edu/entries/logic-manyvalued/ Many-valued logic] " -- by Siegfried Gottwald.