Rough Set Theory and Logic-Algebraic Structures

Any Rough Set System induced by an Approximation Space can be given several logic-algebraic interpretations related to the intuitive reading of the notion of Rough Set. In this paper Rough Set Systems are investigated, first, within the framework of Nelson algebras and the structure of the resulting subclass is inherently described using the properties of Approximation Spaces. In particular, the logic-algebraic structure given to a Rough Set System, understood as a Nelson algebra is equipped with a weak negation and a strong negation and, since it is a finite distributive lattice, it can also be regarded as a Heyting algebra equipped with its own pseudo-complementation. The double weak negation and the double pseudo-complementation are shown to be projection operations connected to the notion of definability in Approximation Spaces. From this analysis we obtain an interpretation of Rough Sets Systems connected to three-valued Lukasiewicz algebras where the roles of projections operators are played by the two endomorphisms of these algebras. Finally, continuing to explore the point of view of Multi-Valued Logics suggested by the latter interpretation we achieve in a quite “natural” way an interpretation based on the notion of Chain Based Lattice. Here the projection operators are provided by the pseudo-supplementation and dual pseudo-supplementation.