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A Paradigm of Granular Computing Based on Quotient Closure Space

A paradigm of granular computing, based on the quotient closure space, is proposed, which is also a generalized version of quotient space theory of problem solving in the sense that closure structure generalizes the classical topological structure. From the point view of granular computing, granulation criteria are equivalence relations and consequently granules are equivalence classes. Granular worlds derived by all the equivalence relations on a given universe, namely granular levels that consist of granules, structure and attribute functions, turn out to be a complete lattice. Both the property preserving, such as pre-order, partial order and connectedness, and the construction of new granular world when given some granular worlds are discussed by means of continuous natural mapping (or, canonical function). The investigation implies that most of the statements of quotient space theory of problem solving still hold in the context of closure operation rather than of Kwiatkowski's closure operator, therefore the content and the applicable domain of the quotient space theory of problem solving are enriched and extended respectively