Rough Sets Through Algebraic Logic

Abstract

While studying rough equality within the framework of the modal system S₅, an algebraic structure called rough algebra [1], came up. Its features were abstracted to yield a topological quasi-Boolean algebra (tqBa). In this paper, it is observed that rough algebra is more structured than a tqBa. Thus, enriching the tqBa with additional axioms, two more structures, viz. pre-rough algebra and rough algebra, are denned. Representation theorems of these algebras are also obtained. Further, the corresponding logical systems ℒ₁ ℒ₂ are proposed and eventually, ℒ₂ is proved to be sound and complete with respect to a rough set semantics.