Six-Set Approximation Theorem of neighborhood related rough sets

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In rough set theory, every rough set can be approximated by lower and upper approximations. Early in Pawlak's classical rough set model, the lower and upper approximations of a set are both crisp sets, but in the generalized rough set models, the lower and upper approximations of a set are generally not crisp sets, i.e., they are still rough sets. This seems very unsatisfactory and deviates from Pawlak's original idea, but we have found the situation is not always so bad. During the study of the theory, a very interesting and important discovery is that, starting from a rough set, after applying the granule-based neighborhood related lower and upper approximation operations to it for any times, we can only get six different sets at most and this six-set family is stable. That is to say, every rough set in a universe can be approximated by only six sets, and the lower and upper approximations of each set in the six-set family are still in the family. We call this result Six-Set Approximation Theorem and the six sets are called Six-Set approximations of the rough set. We study the relationships of the six sets in the family, give some new classification methods of rough sets, and define the stabilization accuracy of each rough set. Moreover, we give the matrix methods to compute the Six-Set approximations and apply them to some well known covering rough set models in literature.