粗糙模糊集的构造与公理化方法

Characterizating Rough Fuzzy Sets in Constructive and Axiomatic Approaches

用构造性方法和公理化研究了粗糙模糊集 .由一个一般的二元经典关系出发构造性地定义了一对对偶的粗糙模糊近似算子 ,讨论了粗糙模糊近似算子的性质 ,并且由各种类型的二元关系通过构造得到了各种类型的粗糙模糊集代数 .在公理化方法中 ,用公理形式定义了粗糙模糊近似算子 ,各种类型的粗糙模糊集代数可以被各种不同的公理集所刻画 .阐明了近似算子的公理集可以保证找到相应的二元经典关系 ,使得由关系通过构造性方法定义的粗糙模糊近似算子恰好就是用公理化定义的近似算子 .

The theory of rough sets can be developed in at least two approaches: constructive and axiomatic approaches, which are complementary to each other. The constructive approach is more suitable for practical applications of rough sets, while the axiomatic approach is appropriate for studying the structures of rough set algebra. In this paper, the constructive and axiomatic approaches in the study of rough fuzzy set are presented. In the constructive approach, one starts from a binary crisp relation and defines a pair of lower and upper rough fuzzy approximation operators. Different classes of rough fuzzy set algebra are obtained from different types of binary crisp relations. In the axiomatic approach, one defines a pair of dual rough fuzzy approximation operators and states that axioms must be satisfied by the operators. Various classes of rough fuzzy algebra are characterized by different sets of axioms. Axioms of rough fuzzy approximation operators guarantee the existence of certain types of binary crisp relations producing the same rough fuzzy operators.