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 (BZMV(dM)-algebra). We study this structure and sketch its main properties. A BZMV(dM)-algebra is a system endowed with a commutative and associative binary operator + and two unusual orthocomplementations: a Kleene orthocomplementation ((__)(\)) and a Brouwerian one (similar to). AS expected, every BZMV(dM)-algebra is both an MV-algebra and a distributive de Morgan BZ-lattice. The set of all similar to-closed elements (which coincides with the set of all +-idempotent elements) tums out to be a Boolean algebra (the Boolean algebra of sharp or crisp elements). By means of (__)(\) and similar to) two modal-like unary operators (v for necessity and mu for possibility) can be introduced in such a way that v(a) (resp., mu(a)) can be regarded as the sharp approximation from the bottom (resp., top) of a. This gives rise to the rough approximation (v(a),mu(a)) of a. Finally, we prove that BZMV(dM)- algebras (which are equationally characterized) are the same as the Stonian MV-algebras and a first representation theorem is proved. (C) 1999 Elsevier Science B.V. All rights reserved.