An MV-algebra is an algebraic structure which models Luzasiewicz multivalued logic, and the fragment of that calculus which deals with the basic logical connectives “and”, “or”, and “not”, but in a multivalued context.
An MV-algebra consists of
a non-empty set, ;
a binary operation, , on ;
a unary operation, , on ; and a constant, , such that
is a commutative monoid;
for all ;
for all ; and
for all in .
The last axiom is more difficult to interpret but is clarified by some examples.
Let be the unit interval. Define
x\oplus y = min(1,x+y)
\neg x = 1-x.
This gives a commutative monoid easily enough and the double negation and absorption axioms are easy to check. Finally the last axiom divides into two cases: and and only one of these needs checking!
Any Boolean algebra defines a MV-algebra with , and being the complement operation. The expressions in the last axiom evaluate to .
Define , and define .
Each MV algebra carries a lattice structure, where the join and meet operations are defined by
x \vee y = (x \odot \neg y) \oplus y
x \wedge y = x \odot (\neg x \oplus y)
and where and are the bottom and top elements.
Let be the partial ordering for the lattice structure, where if .
Each MV algebra is a residuated lattice?, i.e., a closed monoidal poset, where the monoidal product is and
x \odot y \leq z \qquad iff \qquad x \leq y \Rightarrow z.
In particular, for all elements and we have , , and .
In fact, an MV algebra is a -autonomous poset, where .
The equational variety of MV algebras is a Mal'cev variety, where the Mal’cev operation is defined by
t(x, y, z) = ((x \Rightarrow y) \Rightarrow z) \wedge ((z \Rightarrow y) \Rightarrow x).
In other words, .
The variety of MV algebras is generated by with its standard MV algebra structure. Consequently, an identity holds for all MV algebras if and only if it holds in .