MV-algebra
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In [[abstract algebra]], a branch of pure [[mathematics]], an '''MV-algebra''' is an [[algebraic structure]] with a [[binary operation]] <math>\oplus</math>, a [[unary operation]] <math>\neg</math>, and the constant <math>0</math>, satisfying certain axioms. MV-algebras are [[model theory|models]] of [[Łukasiewicz logic]]; the letters MV refer to ''multi-valued'' logic of Łukasiewicz.
==Definitions==
An '''MV-algebra''' is an [[algebraic structure]] <math>\langle A, \oplus, \lnot, 0\rangle,</math> consisting of
* a [[empty set|non-empty]] [[set]] <math>A,</math>
* a [[binary operation]] <math>\oplus</math> on <math>A,</math>
* a [[unary operation]] <math>\lnot</math> on <math>A,</math> and
* a constant <math>0</math> denoting a fixed [[element (mathematics)|element]] of <math>A,</math>
which satisfies the following [[identity (mathematics)|identities]]:
* <math> (x \oplus y) \oplus z = x \oplus (y \oplus z),</math>
* <math> x \oplus 0 = x,</math>
* <math> x \oplus y = y \oplus x,</math>
* <math> \lnot \lnot x = x,</math>
* <math> x \oplus \lnot 0 = \lnot 0,</math> and
* <math> \lnot ( \lnot x \oplus y)\oplus y = \lnot ( \lnot y \oplus x) \oplus x.</math>
By virtue of the first three axioms, <math>\langle A, \oplus, 0 \rangle</math> is a commutative [[monoid]]. Being defined by identities, MV-algebras form a [[variety (universal algebra)|variety]] of algebras. The variety of MV-algebras is a subvariety of the variety of [[BL-algebra]]s and contains all [[Boolean algebra (structure)|Boolean algebra]]s.
An MV-algebra can equivalently be defined (Hájek 1998) as a prelinear commutative bounded integral [[residuated lattice]] <math>\langle L, \wedge, \vee, \otimes, \rightarrow, 0, 1 \rangle </math> satisfying the additional identity <math>x \vee y = (x \rightarrow y) \rightarrow y.</math>
==Examples of MV-algebras==
A simple numerical example is <math>A=[0,1],</math> with operations <math>x \oplus y = \min(x+y,1)</math> and <math>\lnot x=1-x.</math> 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, <math>0\oplus0=0</math> and <math>\lnot0=0.</math>
The ''two-element'' MV-algebra is actually the [[two-element Boolean algebra]] <math>\{0,1\},</math> with <math>\oplus</math> coinciding with Boolean disjunction and <math>\lnot</math> 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 <math>n+1</math> equidistant real numbers between 0 and 1 (both included), that is, the set <math>\{0,1/n,2/n,\dots,1\},</math> which is closed under the operations <math>\oplus</math> and <math>\lnot</math> 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 logic]]s, 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 (mathematics)|valuation]] is a [[homomorphism]] from the algebra of [[propositional formula]]s (in the language consisting of <math>\oplus,\lnot,</math> and 0) into ''A''. Formulas mapped to 1 (or <math>\lnot</math>0) for all ''A''-valuations are called ''A''-[[tautology|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 [[two-element Boolean algebra|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.
[[Category:Fuzzy logic]]
[[Category:Mathematical logic]]
[[zh:MV-代数]]