Abstract: Pawlak proposed the rough set theory in order to process data and knowledge which are imprecise or uncertainty in artificial intelligence. And then, the theory got extended. There're generally two methods; one is to weaken the dependence on equivalence relations,the other is to develop domains to be studied from one to many. Based on the two kinds of thoughts, we researched a product fuzzy rough set model based on two fuzzy approximate spaces, and representation and decomposition of fuzzy rough sets in the product fuzzy approximation spaces. We could explore questions of fuzzy knowledge Granularity's expression from different angles in the high dimension fuzzy knowledge space. We first researched hierarchical structure of a fuzzy approximation space-kcut approximation spaces,and gained the relationship between vary hierarchical knowledge granularity. Secondly, the product of finite fuzzy equivalence relations was defined, and its algorithm was investigated. Finally, a product fuzzy approximation space was constructed based on product fuzzy equivalence relations, and decompositions of upper and lower approximations of fuzzy sets were discussed in the high dimension fuzzy approximation space,and a characterization of upper(lower) approximation of crisp decomposable sets was given.

Key words: cut approximation space,Fuzzy rough set,Product fuzzy cquivalence relation,Product fuzzy approximation space,Decomposable set