Granular Computing on Partitions, Coverings and Neighborhood Systems

Granular Computing on partitions (RST), coverings (GrCC) and neighborhood systems (LNS) are examined: 1. The order of generality is RST, GrCC, and then LNS 2. The quotient structure: In RST, it is called quotient set. In GrCC, it is a simplical complex, called the nerve of the covering in combinatorial topology. For LNS, the structure has no known description. 3. The approximation space of RST is a topological space generated by a partition, called a clopen space. For LNS, it is a generalized/pre topological space which is more general than topological space. For GrCC, there are two possibilities. One is a special case of LNS, which is the topological space generated by the covering. There is another topological space, the topology generated by the finite intersections of the members of a covering The first one treats covering as a base, the second one as a subbase. 4. Knowledge representations in RST are symbol-valued systems. In GrCC, they are expression-valued systems. In LNS, they are multivalued system; reported in 1998l 5. RST and GRCC representation theories are complete in the sense that granular models can be recaptured fully from the knowledge representations. keyword granular computing, neighborhood system, rough set, topology, simplicial complex