On axiomatic characterisations of crisp approximation operators ?
In rough set theory one can find the result that the upper approximation defines a closure operator on the power set of the universe fixed. In the paper presented, first, we have solved the problem under which additional assumptions a given closure operator can be represented by an upper approximation within the concepts of rough set theory. Second, we have solved this problem within modal logic for a diamond operator represented by an arbitrary binary relation on the universe. Finally, we have generalized these problems to lower approximations and modal box operators, to rough approximations of fuzzy sets (rough fuzzy sets), to fuzzy approximations of crisp sets (fuzzy rough sets), and fuzzy approximations of fuzzy sets (`fuzzy' diamond and `fuzzy' box).
closure operators rough fuzzy sets fuzzy rough sets upper approximations of rough sets compactness and strong compactness dualization fuzzy diamond and fuzzy box continuous-valued modal logic