Rough Sets Defined by Multiple Relations

Abstract

We generalize the standard rough set pair induced by an equivalence E on U in such a way that the upper approximation defined by E is replaced by the upper approximations determined by tolerances$$T:{1},\ldots,T_{n}$$ on U. Using this kind of multiple upper approximations we can express “softer” uncertainties of different kinds. We can order the set$$ RS (E,T_{1},\ldots,T_{n})$$ of the multiple approximations of all subsets of the universe U by the coordinatewise inclusion. We show that whenever the tolerances$$T:{1},\ldots,T_{n}$$ are E-compatible, this ordered set forms a complete lattice. As a special case we show how this complete lattice can be reduced to the complete lattice of the traditional rough sets defined by the equivalence E.