L-fuzzy集理论的范畴基础与层表示

鉴于L-fuzzy集在理论上的重要性和应用上的广泛性,旨在建立L-fuzzy集理论的范畴基础与它的层表示,提出完备范畴中对象上的格值结构概念,这一概念是L-fuzzy结构在范畴层面上的提升,进一步提出完备范畴上格值结构提升范畴概念,证明了在集合范畴中L-fuzzy结构与格值结构是同构的.以集层、群层、环层和左R-模层以及Grothendieck层等概念为基础,提出完备范畴中对象上的层结构以及完备范畴上层结构提升范畴概念,证明了在集合范畴中L-fuzzy结构与层结构也是同构的.

The Categorical Foundation of L-fuzzy Sets Theory and Its Sheaves Representation

In view of the universality of L-fuzzy sets theory in application,the aims of this paper are to establish the categorical foundation of L-fuzzy sets,and to present its sheaves representation.The fundamental concept of lattice-valued structure,on a object that is in the complete category,is introduced.This concept is a lifting concept of L-fuzzy structure in category horizon.Further up,this paper puts forward the lifting category of lattice-valued structure on a complete category,and proves isomorphic relation between the L-fuzzy structure and the lattice-valued structure in the category of sets.Based on the concepts of the set sheaf, the group sheaf,the ring sheaf,the left R-module sheaf and Grothendieck sheaf,this paper also puts forward two concepts.One concept is the sheaf structure on a object which is in the complete category.The other concept is the lifting category of sheaf structures on a complete category.In addition,it establishes isomorphic relation between the L-fuzzy structure and sheaf structure in the category of sets.