Rough Approximate Operators: Axiomatic Rough Set Theory
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TL;DR
This paper considers the reverse problem, where H and L are two abstract operators acting on the power set of U, the universe of discourse, and finds that there is an equivalence relation defined on U such that H (X) and L(X) are precisely the upper and lower approximations.
Abstract
In rough set theory, the upper and lower approximations are defined in terms of equivalence relation. In this paper, the reverse problem is considered. Let H and L are two abstract operators acting on the power set of U, the universe of discourse. If the two operators satisfy six axioms, then there is an equivalence relation defined on U such that H(X) and L(X) are precisely the upper and lower approximations. The six axioms are adopted from the axioms of Kuratowski's closure operator. The proof is an easy application of point set topology. Similar results (five axioms) are also obtained for neighborhood systems (a generalized rough set theory) which are based on Frechet (V)spaces. The results can be viewed as a beginning of an axiomatic rough set theory.