BZMV~dM algebras and stoning MV-algebras (applications to fuzzy sets and rough approximations)

摘要

The natural algebraic structure of fuzzy sets suggests the introduction of an abstract algebraic structure called by 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 (direct ＋) and two unusual orthocomplemetnations: a Kleene orthocomplementation (┌ ) and a Brouwerian one (～). As expected, every BZMV~dM-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 (direct ＋ )-diempotent elements) turns out to be a Boolean algebra (the Boolean algebra of sharp or crisp elements). By means of ┌ and ～, two modal-lie unary operators (υ for necessity and μ for possibility) can be introduced in such a way that υ(α) (resp., μ( α)) can be regarded as the sharp approximation form the bottom (resp., top) of α. This gives rise to the rough approximation (υ(α ),μ(α)) of α. Finally, we prove that BZMV~dM-algebras (which are equationally characterized) are the same as the Stoning MV-algebras and a first representation theorem is proved.