MV-algebra

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In mathematics, an MV-algebra an algebraic structure first devised by Jan Łukasiewicz to study multi-valued logic. Chang's completeness theorem (1958, 1959) states that any MV-algebra equation holding over the interval [0,1] will hold in every MV-algebra. Hence MV-algebras characterize infinite-valued Łukasiewicz logics, a fact that extends naturally to fuzzy logic. The way the [0,1] MV-algebra characterizes all possible MV-algebras parallels the well-known fact that the two-element Boolean algebra (with carrier {0,1}) characterizes all possible Boolean algebras. Moreover, the way MV-algebras characterize infinite-valued logics is analogous to the way that Boolean algebras characterize standard bivalent (two valued) logic.

1 Definitions
2 Applications
3 References
4 External links

[edit] Definitions

Let A be some underlying set. An MV-algebra is a $\left \langle A, \oplus, \lnot, 0 \right \rangle$ algebra, such that $\left \langle A, \oplus, 0 \right \rangle$ is a commutative monoid satisfying the additional identities:

$\lnot \lnot x = x$ ,
$x \oplus \lnot 0 = \lnot 0$ , and
$\ \lnot ( \lnot x \oplus y)\oplus y = \lnot ( \lnot y \oplus x) \oplus x$ .

An MV-algebra may also be defined as a residuated lattice $A= \left \langle L, \wedge, \vee, \otimes, \rightarrow, 0, 1 \right \rangle$ satisfying the additional identity $x \vee y = (x \rightarrow y) \rightarrow y \$ .

On the equivalence between these two formulations, see Hájek (1998).

[edit] Applications

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

Given some MV-algebra A, an A-valuation is a function from the set of propositional logic formulas into A. Formulas mapped to 1 (or $\lnot$ 0) for all A-valuations are A-tautologies. Thus for infinite-valued logics (i.e. fuzzy logic, Łukasiewicz logic), we let [0,1] be the underlying set of A to obtain [0,1]-valuations and [0,1]-tautologies (often simply called "valuations" and "tautologies").

[edit] References

Chang, and Keisler, J., 1973. Model Theory. North Holland.
Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., 2000. Algebraic Foundations of Many-valued Reasoning. Kluwer.
Di Nola A. , Lettieri A. , Equational characterization of all varieties of MV-algebras, Journal of Algebra 221 (1993) 123-131.
Petr Hájek, 1998. Metamathematics of Fuzzy Logic. Kluwer.