Granular models for partitions, coverings, and Pre-Topologies

Abstract

Granular computing on partitions (RST), coverings (GrCC) and pre-topology (neighborhood systems (NS)) are examined: 1) the order of generality is RST, GrCC, and then NS. 2) The quotient structures: In RST, it is the quotient set. In GrCC, it is a simplicial complex (in combinatorial topology). For NS, the structure has no known description. 3) A new and traditional approximation theories are presented: For both theories, RST is a topological space generated by a partition, called a clopen space. For NS, the classical one is a pre-topological space, a generalized topological spaces. For new one, it is a topological space. For GrCC, there are two possibilities. One is a special case of NS, which is the topological space generated by the covering. There is another topological space, the topology is 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 NS, they are multi-valued system. 5) For RST, GrCC and Symmetric NS, the representation theories are complete in the sense that granular models can be recaptured fully by knowledge representations.