Singly and doubly even

In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation $\oplus$, a unary operation $\mathrm{\neg }$, and the constant $0$, satisfying certain axioms. MV-algebras are the algebraic semantics of Łukasiewicz logic; the letters MV refer to many-valued logic of Łukasiewicz. MV-algebras coincide with the class of bounded commutative BCK algebras.

Definitions

An MV-algebra is an algebraic structure $⟨A,\oplus ,\mathrm{\neg },0⟩,$ consisting of

• a non-empty set $A,$
• a binary operation $\oplus$ on $A,$
• a unary operation $\mathrm{\neg }$ on $A,$ and
• a constant $0$ denoting a fixed element of $A,$

which satisfies the following identities:

• $(x\oplus y)\oplus z=x\oplus (y\oplus z),$
• $x\oplus 0=x,$
• $x\oplus y=y\oplus x,$
• $\mathrm{\neg }\mathrm{\neg }x=x,$
• $x\oplus \mathrm{\neg }0=\mathrm{\neg }0,$ and
• $\mathrm{\neg }(\mathrm{\neg }x\oplus y)\oplus y=\mathrm{\neg }(\mathrm{\neg }y\oplus x)\oplus x.$

By virtue of the first three axioms, $⟨A,\oplus ,0⟩$ 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 $⟨L,\wedge ,\vee ,\otimes ,\to ,0,1⟩$ satisfying the additional identity $x\vee y=(x\to y)\to y.$

Examples of MV-algebras

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

The two-element MV-algebra is actually the two-element Boolean algebra $\{0,1\},$ with $\oplus$ coinciding with Boolean disjunction and $\mathrm{\neg }$ with Boolean negation. In fact adding the axiom $x\oplus x=x$ to the axioms defining an MV-algebra results in an axiomantization of Boolean algebras.

If instead the axiom added is $x\oplus x\oplus x=x\oplus x$, then the axioms define the MV₃ algebra corresponding to the three-valued Łukasiewicz logic Ł₃Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. Other finite linearly ordered MV-algebras are obtained by restricting the universe and operations of the standard MV-algebra to the set of $n$ equidistant real numbers between 0 and 1 (both included), that is, the set $\{0,1/(n-1),2/(n-1),\dots ,1\},$ which is closed under the operations $\oplus$ and $\mathrm{\neg }$ 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 $\oplus ,\mathrm{\neg },$ and 0) into A. Formulas mapped to 1 (or $\mathrm{\neg }$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).

Relation to functional analysis

Template:Expand section MV-algebras were related by D. 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 }	Mn(ℂ), i.e. n×n complex matrices
finite	finite-dimensional
boolean	commutative

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 a MV-algebra. More information available at Multi-adjoint logic programming.

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.
• Mundici, D.: Interpretation of AF C*-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986) 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.

Further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Stanford Encyclopedia of Philosophy: "Many-valued logic" -- by Siegfried Gottwald.