The Investigation On Representation Theorem Of Fuzzy Sets And Its Applications

Firstly, this paper discusses the representation theorems which based onunion-intersection-preserving mapping and intersection-union-preserving map-ping respectively, under the condition of which the membership degree's lattice isonly complete lattice. Secondly, it uses this new representation theorem to solvesome public problems concerning with cuts.Chapter 1 is exordiums. It mainly includes the summarization of this pa-per,the main work and some basic notations.Chapter 2 is the investigation of representation theorems which respec-tively based on union-intersection-preserving mapping and intersection-union-preserving mapping. Representation theorem is one of the three basic theoremsin fuzzy sets theory, it revels the relation between fuzzy sets and classical sets.Hence, lots of experts and scholars have been investigated it intensively, andthe methods and forms are also variety. Whereas, what worth to say is thatthe representation theorem in most of literatures is obtained by strengthen thecondition of the membership degree's lattice. Such as it needs the membershipdegree's lattice was dense complete lattice, completely distribute lattice or unitinterval, ect. Obviously, we expect to weaken the condition of the membershipdegree's lattice as possible as we can. For years, some scientific research workare in developing. But there is one thing we can come to understanding, the re-sult is more significant when the membership degree's lattice is complete lattice.Presently, the work in this respect can be find in literatures. In this chapter,it investigates a new representative theorem under the condition of which themembership degree's lattice is only complete lattice.Chapter 3 is the application of this new representation theorem in solving some public problems concerning with cuts. During the establishment of fuzzysets theory, there are lots of public problems being posed. Some of them are aboutthe relation between fuzzy sets and classical sets. Fuzzy sets is concerned withenvironment of the membership degree's lattice, therefore, the methods of solvingproblems are di?erent when the environment di?erent. In this chapter, under theenvironment of complete lattice, it gives some new solving methods of the threepublic problems. The tool is the representation theorem which based on union-intersection-preserving mapping and intersection-union-preserving mapping.