Partial approximative set theory: A generalization of the rough set theory

There are close links between mathematical morphology and rough set theory. Both theories are successfully applied among others to image processing and pattern recognition. This paper presents a new generalization of the classical rough set theory, called the partial approximative set theory (PAST). According to Pawlak's classic rough set theory, the vagueness of a subset of a finite universe is defined by the difference of its upper and lower approximations with respect to an equivalence relation on the universe. There are two most natural ways of the generalization of this idea. Namely, the equivalence relation is replaced by either any other type of binary relations on the universe or an arbitrary covering of the universe. In this paper, our starting point will be an arbitrary family of subsets of an arbitrary universe, neither that it covers the universe nor that the universe is finite will be assumed. We will give some reasons why this new approach is worth studying, and put our discussions into an overall treatment, called the general approximation framework.