MV-algebra
In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation , a unary operation , and the constant , 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 consisting of
- a non-empty set
- a binary operation on
- a unary operation on and
- a constant denoting a fixed element of
which satisfies the following identities:
By virtue of the first three axioms, 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 satisfying the additional identity
Examples of MV-algebras
A simple numerical example is with operations and 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, and
The two-element MV-algebra is actually the two-element Boolean algebra with coinciding with Boolean disjunction and with Boolean negation. In fact adding the axiom to the axioms defining an MV-algebra results in an axiomantization of Boolean algebras.
If instead the axiom added is , then the axioms define the MV_{3} algebra corresponding to the three-valued Łukasiewicz logic Ł_{3}{{ 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 equidistant real numbers between 0 and 1 (both included), that is, the set which is closed under the operations and of the standard MV-algebra; these algebras are usually denoted MV_{n}.
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 { x ∈ G | 0 ≤ x ≤ u }, which becomes an MV-algebra with x ⊕ y = min(u, x+y) and ¬x = u−x. 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" { x ∈ G | 0 ≤ x ≤ u } can be equipped with ¬x = u−x, x ⊕ y = u∧_{G} (x+y), x ⊗ y = 0∨_{G}(x+y−u). 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 and 0) into A. Formulas mapped to 1 (or 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]}
MV_{n}-algebras
Template:Expand section In the 1940s Grigore Moisil introduced his Łukasiewicz–Moisil algebras (LM_{n}-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 ℵ_{0}-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 MV_{n}-algebras.^{[2]} MV_{n}-algebras are a subclass of LM_{n}-algebras; the inclusion is strict for n ≥ 5.^{[3]}
The MV_{n}-algebras are MV-algebras which satisfy some additional axioms, just like the n-valued Łukasiewicz logics have additional axioms added to the ℵ_{0}-valued logic.
In 1982 Roberto Cignoli published some additional constraints that added to LM_{n}-algebras are proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper n-valued Łukasiewicz algebras.^{[4]} The LM_{n}-algebras that are also MV_{n}-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} | ℂ |
{0, 1/n, ..., 1 } | M_{n}(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
- ↑ 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
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Iorgulescu, A.: Connections between MV_{n}-algebras and n-valued Łukasiewicz–Moisil algebras—I. Discrete Math. 181, 155–177 (1998) Template:Hide in printTemplate:Only in print
- ↑ 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
- ↑ http://journal.univagora.ro/download/pdf/28.pdf
- 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.
- Mundici, D.: Interpretation of AF C*-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986) Template:Hide in printTemplate:Only in print
Further reading
- Daniele Mundici, MV-ALGEBRAS. A short tutorial
- {{#invoke:citation/CS1|citation
|CitationClass=book }}
- Mundici, D. The C*-Algebras of Three-Valued Logic. Logic Colloquium ’88, Proceedings of the Colloquium held in Padova 61–77 (1989). Template:Hide in printTemplate:Only in print
- Cabrer, L. M. & Mundici, D. A Stone-Weierstrass theorem for MV-algebras and unital ℓ-groups. Journal of Logic and Computation (2014). Template:Hide in printTemplate:Only in print
- Olivia Caramello, Anna Carla Russo (2014) The Morita-equivalence between MV-algebras and abelian ℓ-groups with strong unit