In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, a branch of pure
mathematics, an MV-algebra is an
algebraic structure with a
binary operation , a
unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
, and the constant
, satisfying certain axioms. MV-algebras are the
algebraic semantics of
Łukasiewicz logic
In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic;Łukasiewicz J., 1920, O logice trójwartościowej (in P ...
; the letters MV refer to the
''many-valued'' logic of
Łukasiewicz. MV-algebras coincide with the class of bounded
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
BCK algebras.
Definitions
An MV-algebra is an
algebraic structure consisting of
* a
non-empty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
* a
binary operation on
* a
unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
on
and
* a constant
denoting a fixed
element of
which satisfies the following
identities:
*
*
*
*
*
and
*
By virtue of the first three axioms,
is a commutative
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
. Being defined by identities, MV-algebras form a
variety
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
of algebras. The variety of MV-algebras is a subvariety of the variety of
BL-algebras and contains all
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
s.
An MV-algebra can equivalently be defined (
Hájek 1998) as a prelinear commutative bounded integral
residuated lattice In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice ''x'' ≤ ''y'' and a monoid ''x''•''y'' which admits operations ''x''\''z'' and ''z''/''y'', loosely analogous to division or implication, when ' ...
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
In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic;Łukasiewicz J., 1920, O logice trójwartościowej (in P ...
.
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
In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose ''underlying set'' (or universe or ''carrier'') ''B'' is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that ''B ...
with
coinciding with Boolean disjunction and
with Boolean negation. In fact adding the axiom
to the axioms defining an MV-algebra results in an axiomatization 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. 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
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...
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
, ''u''as , 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.
Daniele Mundici extended the above construction to abelian
lattice-ordered group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' t ...
s. If ''G'' is such a group with strong (order) unit ''u'', then the "unit interval" can be equipped with ¬''x'' = ''u'' − ''x'', ''x'' ⊕ ''y'' = ''u'' ∧
G (x + y), and ''x'' ⊗ ''y'' = 0 ∨
G (''x'' + ''y'' − ''u''). This construction establishes a
categorical equivalence between lattice-ordered abelian groups with strong unit and MV-algebras.
An
effect algebra that is lattice-ordered and has the
Riesz decomposition property is an MV-algebra. Conversely, any MV-algebra is a lattice-ordered effect algebra with the Riesz decomposition property.
Relation to Łukasiewicz logic
C. C. Chang devised MV-algebras to study
many-valued logic
Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false ...
s, introduced by
Jan Łukasiewicz
Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic His work centred on philosophical logic, mathematical logic and history of logic. ...
in 1920. In particular, MV-algebras form the
algebraic semantics of
Łukasiewicz logic
In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic;Łukasiewicz J., 1920, O logice trójwartościowej (in P ...
, as described below.
Given an MV-algebra ''A'', an ''A''-
valuation is a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from the algebra of
propositional formula
In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional for ...
s (in the language consisting of
and 0) into ''A''. Formulas mapped to 1 (that is, to
0) for all ''A''-valuations are called ''A''-
tautologies. If the standard MV-algebra over
,1is employed, the set of all
,1tautologies determines so-called infinite-valued
Łukasiewicz logic
In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic;Łukasiewicz J., 1920, O logice trójwartościowej (in P ...
.
Chang's (1958, 1959) completeness theorem states that any MV-algebra equation holding in the standard MV-algebra over the interval
,1will 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
In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic;Łukasiewicz J., 1920, O logice trójwartościowej (in P ...
, defined as the set of
,1tautologies.
The way the
,1MV-algebra characterizes all possible MV-algebras parallels the well-known fact that identities holding in the
two-element Boolean algebra
In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose ''underlying set'' (or universe or ''carrier'') ''B'' is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that ''B ...
hold in all possible Boolean algebras. Moreover, MV-algebras characterize infinite-valued
Łukasiewicz logic
In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic;Łukasiewicz J., 1920, O logice trójwartościowej (in P ...
in a manner analogous to the way that
Boolean algebras
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a gen ...
characterize classical
bivalent logic
In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is calle ...
(see
Lindenbaum–Tarski algebra
In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory ''T'' consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that ''p'' ...
).
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 term-equivalent.
MV''n''-algebras
In the 1940s
Grigore Moisil
Grigore Constantin Moisil (; 10 January 1906 – 21 May 1973) was a Romanian mathematician, computer pioneer, and titular member of the Romanian Academy. His research was mainly in the fields of mathematical logic ( Łukasiewicz–Moisil algebra ...
introduced his
Łukasiewicz–Moisil algebra Łukasiewicz–Moisil algebras (LM''n'' algebras) were introduced in the 1940s by Grigore Moisil (initially under the name of Łukasiewicz algebras) in the hope of giving algebraic semantics for the ''n''-valued Łukasiewicz logic. However, in 195 ...
s (LM
''n''-algebras) in the hope of giving
algebraic semantics for the (finitely) ''n''-valued
Łukasiewicz logic
In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic;Łukasiewicz J., 1920, O logice trójwartościowej (in P ...
. However, in 1956 Alan Rose discovered that for ''n'' ≥ 5, the Łukasiewicz–Moisil algebra does not
model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
the Łukasiewicz ''n''-valued logic. Although C. C. Chang published his MV-algebra in 1958, it is a 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.
MV
''n''-algebras are a subclass of LM
''n''-algebras; the inclusion is strict for ''n'' ≥ 5.
The MV
''n''-algebras are MV-algebras that 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 yield proper models for ''n''-valued Łukasiewicz logic; Cignoli called his discovery ''proper n-valued Łukasiewicz algebras''. The LM
''n''-algebras that are also MV
''n''-algebras are precisely Cignoli’s proper ''n''-valued Łukasiewicz algebras.
Relation to functional analysis
MV-algebras were related by
Daniele Mundici to
approximately finite-dimensional C*-algebra In mathematics, an approximately finite-dimensional (AF) C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by ...
s by establishing a bijective correspondence between all isomorphism classes of approximately finite-dimensional C*-algebras with lattice-ordered dimension group and all isomorphism classes of countable MV algebras. Some instances of this correspondence include:
In software
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 an MV-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: 463–474 .
* 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)
Further reading
* Daniele Mundici
MV-ALGEBRAS. A short tutorial*
* Mundici, D. The C*-Algebras of Three-Valued Logic. Logic Colloquium ’88, Proceedings of the Colloquium held in Padova 61–77 (1989).
* Cabrer, L. M. & Mundici, D. A Stone-Weierstrass theorem for MV-algebras and unital ℓ-groups. Journal of Logic and Computation (2014). {{doi, 10.1093/logcom/exu023
*
Olivia Caramello, Anna Carla Russo (2014
The Morita-equivalence between MV-algebras and abelian ℓ-groups with strong unit
External links
*
Stanford Encyclopedia of Philosophy:
Many-valued logic—by
Siegfried Gottwald
Siegfried Johannes Gottwald (30 March 1943 – 20 September 2015) was a German mathematician, logician and historian of science.
Life and work
Gottwald was born in Limbach, Saxony in 1943. From 1961 to 1966, he studied mathematics at the Unive ...
.
Algebraic logic
Algebraic structures
Fuzzy logic
Many-valued logic