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In abstract algebra, an MV-algebra is a set with a number of operations that satisfies certain axioms, detailed below. The concept was first devised by Chang to study structures introduced by Jan Łukasiewicz as 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.

Definitions

An MV-algebra is a set with three operations: one binary operation: , one unary operation: , and one nullary operation (or constant): . So it is a mathematical structure of the form , which satisfys the following identities:
  • ,
  • ,
  • ,
  • ,
  • , and
  • .
Note that the first three axioms simply say that is a commutative monoid.

An MV-algebra may also be defined as a residuated lattice satisfying the additional identity .

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

Applications

A simple numerical example is with operations and .

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 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").

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.

External links

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Most authors nowadays simply write algebra instead of abstract algebra.
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SET may stand for:
  • Sanlih Entertainment Television, a television channel in Taiwan
  • Secure electronic transaction, a protocol used for credit card processing,

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In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary and binary.
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Multi-valued logics are logical calculi in which there are more than two truth values. Traditionally, logical calculi are two-valued—that is, there are only two possible truth values (i.e. truth and falsehood) for any proposition to take.
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Multi-valued logics are logical calculi in which there are more than two truth values. Traditionally, logical calculi are two-valued—that is, there are only two possible truth values (i.e. truth and falsehood) for any proposition to take.
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Fuzzy Logic may refer to:
  • Fuzzy Logic (album), the debut album by the Super Furry Animals
  • Fuzzy logic, an application of fuzzy set theory
For the music album, see Fuzzy Logic (album)


Fuzzy logic
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The two-element Boolean algebra is the simplest Boolean algebra, one having just two elements, named 1 and 0 by convention. Paul Halmos's name for this algebra, 2, has some following in the literature and will be employed here.
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Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations.
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The two-element Boolean algebra is the simplest Boolean algebra, one having just two elements, named 1 and 0 by convention. Paul Halmos's name for this algebra, 2, has some following in the literature and will be employed here.
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In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator.
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In mathematics, a unary operation is an operation with only one operand, i.e. an operation with a single input, or in other words, a function of one variable (for the terminology see also operators versus functions).
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In logic, mathematics, and computer science, the arity (synonyms include type, adicity, and rank) of a function or operation is the number of arguments or operands that the function takes.
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In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance.
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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. Monoids occur in a number of branches of mathematics.
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In abstract algebra, a residuated lattice is an algebraic structure that is both a lattice and a monoid, such that each argument of the latter's multiplication is a polarity of a Galois connection with respect to the lattice order.
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Informally, a valuation is an assignment of particular values to the variables in a mathematical statement or equation.

In logic and model theory, a valuation
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In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules
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Fuzzy Logic may refer to:
  • Fuzzy Logic (album), the debut album by the Super Furry Animals
  • Fuzzy logic, an application of fuzzy set theory
For the music album, see Fuzzy Logic (album)


Fuzzy logic
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Multi-valued logics are logical calculi in which there are more than two truth values. Traditionally, logical calculi are two-valued—that is, there are only two possible truth values (i.e. truth and falsehood) for any proposition to take.
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The Stanford Encyclopedia of Philosophy (SEP) is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. The SEP was initially developed with U.S. public funding from the NEH and NSF.
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