Study on Fuzzy Granular Space Based on Normalized Equicrural Metric

In this paper, by introducing the normalized equierural metric into fuzzy quotient space theory, the fuzzy granular space theory based on normalized equicrural metric is presented and four main results are obtained. Firstly, the following three statements are equivalent (Theorem3.2): (1) give a fuzzy equivalence relations set on X; (2) give a normalized equicrural metric on X; (3) give an hierarchical structure (or order-complete granular space) on X. Secondly, the correspondence relations between a normalized equicrural metric on X and equivalence relations on universe X are discussed, and they are one-to-many (Theorem2.2, Theorem2.3). Finally, for given a fuzzy equivalence relation on X, the metric d on its deriving fuzzy granulation is determined by its deriving metric, where it just is the reducing distance by d on the granulation. These results provide a powerful mathematics model and tool for granular computing and its application, further provide a direct and geometric explain for granular computing, and will help us for deeply understanding the essence of granular procedure.