On the structure of the multigranulation rough set model

The original rough set model, i.e., Pawlak's single-granulation rough set model has been extended to a multigranulation rough set model, where two kinds of multigranulation approximations, i.e., the optimistic and pessimistic approximations were introduced. In this paper, we consider some fundamental properties of the multigranulation rough set model, and show that (i)Both the collection of lower definable sets and that of upper definable sets in the optimistic multigranulation rough set model can form a lattice, such lattices are not distributive, not complemented and pseudo-complemented in the general case. The collection of definable sets in the optimistic multigranulation rough set model does not even form a lattice in general conditions. (ii)The collection of (lower, upper) definable sets in the optimistic multigranulation rough set model forms a topology on the universe if and only the optimistic multigranulation rough set model is equivalent to Pawlak's single-granulation rough set model. (iii)In the context of the pessimistic multigranulation rough set model, the collections of three different kinds of definable sets coincide with each other, and they determine a clopen topology on the universe, furthermore, they form a Boolean algebra under the usual set-theoretic operations.