On Characterizing Hierarchies of Granulation Structures via Distances

Hierarchy plays a crucial role in the development of the granular computing. In this paper, three different hierarchies are considered for judging whether a granulation structure is finer or coarser than another one. The first hierarchy is based on the set containment of information granulations, the second hierarchy is based on the cardinal numbers of information granulations while the third hierarchy is based on the sum of cardinal numbers of information granulations. Through introducing set distance and knowledge distance, we investigate the algebraic lattices, in which the derived partial orders are corresponding to the three different hierarchies, respectively. From the viewpoint of distance, these results look forward to provide a more comprehensible perspective for the study of hierarchies on granulation structures.