A UNIFIED FRAMEWORK FOR THE ALGEBRA OF UNSHARP QUANTUM-MECHANICS

On the basis of the concrete operations definable on the set of effect operators on a Hilbert space, an abstract algebraic structure of sum Brouwer-Zadeh (SBZ)algebra is introduced. This structure consists of a partial sum operation and two mappings which turn out to be Kleene an d Brouwer unusual orthocomplementations. The Foulis-Bennett effect alg ebra substructure induced by any SBZ-algebra, allows one to introduce the notions of unsharp ''state'' and ''observable'' in such a way that any ''state-observable'' composition is a standard probability measur e (classical state). The Cattaneo-Nistico BZ substructure induced by a ny SBZ-algebra permits one to distinguish, in an equational and simple way, the sharp elements from the really unsharp ones. The family of a ll sharp elements turns out to be a Foulis-Randall orthoalgebra. Any u nsharp element can be ''roughly'' approximated by a pair of sharp elem ents representing the best sharp approximation from the bottom and fro m the top respectively, according to an abstract generalization introd uced by Cattaneo of Pawlack ''rough set'' theory (a generalization of set theory, complementary to fuzzy set theory, which describes approxi mate knowledge with applications in computer sciences). In both the co ncrete examples of fuzzy sets and effect operators the ''algebra'' of rough elements shows a weak SBZ structure (weak effect algebra plus BZ standard poset) whose investigation is set as an interesting open pro blem.