Hierarchical Structures on Multigranulation Spaces

Though many hierarchical structures have been proposed to analyze the finer or coarser relationships between two granulation spaces, these structures can only be used to compare the single granulation spaces. However, it should be noticed that the concept of multigranulation plays a fundamental role in the development of granular computing. Therefore, the comparison between two multigranulation spaces has become a necessity. To solve such problem, two types of the multigranulation spaces are considered: one is the partition-based multigranulation space, the other is the covering-based multigranulation space. Three different hierarchical structures are then proposed on such two multigranulation spaces, respectively. Not only the properties about these hierarchical structures are discussed, but also the relationships between these hierarchical structures and the multigranulation rough sets are deeply investigated. It is shown that the first hierarchical structure is consistent with the monotonic varieties of optimistic multigranulation rough set, and the second hierarchical structure is consistent to the monotonic varieties of pessimistic multigranulation rough set, the third hierarchical structure is consistent to the monotonic varieties of both optimistic and pessimistic multigranulation rough sets.