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
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
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
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
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 .