Research Of Level Sets And Representation Of L-subsets And L-families

In this paper, the purpose is triple. Firstly, the paper discusses the new representation theorem of L—subsets based on L—nested systems under the con-dition of complete residuated lattice. Secondly, with the tool of the system of L—subsets, the representation theorem of intersection-preserving L—families on the condition of complete residuated lattice are obtained. Finally, representation of L—filters and L—generalized convergence structures are discussed under the condition of Heyting algebra.Chapter 1 is exordiums and preliminaries. It mainly includes the summa-rization of this paper, the advance of problems, the main work of this paper and preliminaries. The background and current situation about the research of level sets and representation theorems of L—subsets are introduced, and the main problems to be solved are proposed and the preliminaries used in this paper are given.Chapter 2 is the investigation of representation theorem of L—subsets on the condition of complete residuated lattice. Representation theorem is one of the three basic theorems in fuzzy sets theory, it revels the relation between L—subsets and classical sets. Hence, lots of experts and scholars have investigated it, and the methods and forms are also variety. The representation theorems in a great amount of literature are obtained under the conditions of dense complete lattice, completely distributive lattice, unit interval, complete lattice and so on. In order to describe problems with L—subsets on the condition of lattice-valued logic, scholars usually use complete residuated lattice as the membership degree value lattice of L—subsets. In this chapter, the tensor and residuum operations on L—sested systems are introduced under the condition of complete residuated lattice. Then it is shown that L—nested systems form a complete residuated lattice, which is precisely the classical isomorphic object of complete residuated power set lattice. Thus the new representation theorem of L—subsets on the condition of complete residuated lattice is obtained.Chapter 3 focus on the investigation of level sets and representation theorem of L—families. In scholars'investigations, some special L—subsets which are mappings from Lx to L are common objects, such as many-valued filters, lattice-valued convergence structures and so on, which will be called L—families. In order to research L—families from the classical point of view, this chapter introduces the concept of the system of L—subsets on the condition of complete residuated lattice. Moreover, with the tool of the system of L—subsets, the representation theorem of intersection-preserving L—families under the condition of complete residuated lattice are obtained.Chapter 4 discusses representation of L—fiters and L—generalized conver-gence structures. As L—filters and L—generalized convergence structures can be seen as special L—subsets, with the construction thought of L—nested systems, pre-filter nested systems and limit nested systems are constructed. As main tools of level analysis, pre-filter nested systems and limit nested systems are used to es-tablish the representation theorems of L—filters and L—generalized convergence structures.