The Fuzzy Category Based On The Minimal Extension Principle

Zadeh's extension principle provides a mathematical approach for extending classical functions to fuzzy mapings. It has been considered that extension principle is an important tool in the development of fuzzy arithmetic and other areas. In some cases, the extension principle may be not appropriate for pessimistic or conservative decision. So we proposed the new extension principle—Minimal extension principle. In this paper, we first defined strong(-|α)-lower cut set and weak (-|α)-lower cut set. And discuss the properties theyhave. Because of the defining of that two new cut sets, we can get this conclusion that the two fuzzy sets are equal if and only if the two fuzzy set' s strong(-|α)-lower cut sets are equal. In this paper, most of the proofs used this method.Using Zadeh' s extension principle, we can extend the category C ' s three functions —dom,cod and I to dont, cod and I.So we get the fuzzy categorybased on the Zadeh' s extension principle. In this paper that three functions of category Care extended by using the minimal extension principle. In this way we can define the fuzzy category based on the minimal extension principle. And then we discuss the morphism, object, fuzzy domain and fuzzy codomain's definition and their properties. At the end of this paper, we propose the definition of fuzzy functor.