An MV-algebra is an algebraic structure consisting of
- a non-empty set
- a binary operation on
- a unary operation on and
- a constant denoting a fixed element of
which satisfies the following identities:
By virtue of the first three axioms, 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 satisfying the additional identity
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... 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 ...