MV-algebras coincide with the class of bounded commutative BCK algebras.
An MV-algebra can equivalently be defined (Hájek 1998) as a prelinear commutative bounded integral residuated lattice
to the axioms defining an MV-algebra results in an axiomatization of Boolean algebras.
, then the axioms define the MV3 algebra corresponding to the three-valued Łukasiewicz logic Ł3[citation needed].
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.
Daniele Mundici extended the above construction to abelian lattice-ordered groups.
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.
[1] C. C. Chang devised MV-algebras to study many-valued logics, introduced by Jan Łukasiewicz in 1920.
Given an MV-algebra A, an A-valuation is a homomorphism from the algebra of propositional formulas (in the language consisting of
If the standard MV-algebra over [0,1] is employed, the set of all [0,1]-tautologies determines so-called infinite-valued Łukasiewicz logic.
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.
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 a faithful model only for the ℵ0-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 MVn-algebras.
In 1982, Roberto Cignoli published some additional constraints that added to LMn-algebras yield proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper n-valued Łukasiewicz algebras.
[5] The LMn-algebras that are also MVn-algebras are precisely Cignoli's proper n-valued Łukasiewicz algebras.
[6] MV-algebras were related by Daniele Mundici to approximately finite-dimensional C*-algebras 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.