Structures Derived from Possible Tables in an Incomplete Information Table

We describe structures derived from possible tables when we obtain rough sets in an incomplete information table. An incomplete information table is replaced by the set of possible tables derived from it under possible world semantics. One possible indiscernibility relation is derived from one possible table. We show that the family that consists of the possible indiscernibility relations, their intersection, and their union has a lattice structure. Indiscernible classes of each object, a covering, lower and upper approximations, and a set of rules are obtained from a possible indiscernibility relation. These families also have lattice structures. Under the lattice structures, the lower and upper bounds of rough sets are derived using only the minimum and maximum elements. Consequently, the approach based on possible world semantics has no difficulty with computational complexity for the number of objects with incomplete information.