Representation Theorem Of L-subsets On Lattice-valued Partially Ordered Set

This paper focuses on the representation theorem of L-subsets, where the underlying lattice L is a complete residuated lattice lattice and the L-subset is defined on an L-partially ordered set (X, P). Firstly, the paper discusses the new representation theorem of intersection-preserving L-subsets by using union-preserving system of elements. Secondly, based on the domain endowed with L-partial order, we introduced so-called compatible L-subsets and compatible union-preserving system of elements which cohere with the L-partial order. The representation-preserving compatible L-subset are obtained with the tool of the compatible union-preserving system of elements. Finally, the representation theorem of compatible L-subsets are discussed based on compatible L-nested systems and compatible strong L-nested systems.Chapter 1 is exordiums. It mainly includes the summarization of this paper, the advance of problems and the main work of this paper. 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 in this paper are given.Chapter 2 is preliminaries. This chapter mainly introduce the used sign and symbol. And We have briefly reviewed some basic knowledge such as lattice, order, L-subsets, L-partially ordered set, L-nested systems and so on.Chapter 3 is the investigation of representation theorem of L-subsets on lattice-valued partially ordered set. 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. With the development of representation theorem of L-subsets, it is value to note that these representation theorems of L-subsets have a common characteristic that domain X is just a set, and don’t have any other mathematic structure. But in many areas of fuzzy mathematics, the domain X is often endowed with L-partial order P. Therefore, in this chapter, we proposed the concept of level-elements and union-preserving system of elements. The representation theorem of intersection-preserving L-subsets is successfully obtain by means of union-preserving system of elements.Chapter 4 is the investigation of representation theorem of so-called compati-ble L-subsets which are in accordance with lattice-valued partial order. In schol-ars’investigations, some special L-subsets which are mapping from (X, P) to L are common objects, such as stratified L-filter, which will be called compatible L-subsets. In order to research it from the classical point of view, this chap-ter introduces the concept of compatible union-preserving system of elements. Moreover, with the tool of compatible union-preserving system of elements, the representation theorem of compatible intersection-preserving L-subsets is ob-tained.Chapter 5 is the investigation of representation theorem compatible L-subsets based on the compatible L-nested systems and compatible strong L-nested systems. There is a one-to-one correspondence relation between L-subsets and L-nested systems, strong L-nested systems. Therefore, based on the theory of domain endowed with lattice-valued partial order structure, we introduce the con-cept of compatible L-nested systems and compatible strong L-nested systems. With the tool of compatible L-nested systems and compatible strong L-nested systems, the representation theorem of compatible L-subsets is obtained.