BZMVdM algebras and stonian MV-algebras (applications to fuzzy sets and rough approximations)

The natural algebraic structure of fuzzy sets suggests the introduction of an abstract algebraic structure called de Morgan BZMV-algebra (BZMVdM-algebra). We study this structure and sketch its main properties. A BZMVdM-algebra is a system endowed with a commutative and associative binary operator ⊖ and two unusual orthocomplementations: a Kleene orthocomplementation (¬) and a Brouwerian one (∼). As expected, every BZMVdM-algebra is both an MV-algebra and a distributive de Morgan BZ-lattice. The set of all ∼-closed elements (which coincides with the set of all ⊕-idempotent elements) turns out to be a Boolean algebra (the Boolean algebra of sharp or crisp elements). By means of ¬ and ∼, two modal-like unary operators (ν for necessity and μ for possibility) can be introduced in such a way that ν(a) (resp., μ(a)) can be regarded as the sharp approximation from the bottom (resp., top) of a. This gives rise to the rough approximation (ν(a), μ(a)) of a. Finally, we prove that BZMVdM-algebras (which are equationally characterized) are the same as the Stonian MV-algebras and a first representation theorem is proved