Fuzzy Co-quantales And Fuzzy C-continuous Lattices

Abstract The concepts of fuzzy posets are considered as generalizations of the concepts of classic posets. In the classic poset theory, the simply order relations lack the quantitative information which is needed during the computing. That is, it can not reflect the amount of information which can be computed contained in the elements. The introduction of the fuzzy posets makes up this shortfall. Thus, the study of the quantitative domain theory has aroused interests of many scholars. The first part of this thesis is to study the fuzzification of co-quantale theory. The second part is to introduce the concept of fuzzy C-continuous lattice and study some properties of it. The structure of this thesis is organized as follows:Chapter One:Preliminary knowledge. In this chapter, we give some basic concepts and results that related with the co-quantale theory, the category theory and the fuzzy poset theory which will be used throughout the thesis.Chapter Two:Nuclei and conuclei on co-quantales. Firstly, some properties of nuclei and conuclei on co-quantale are investigated. Secondly, the relationships among Dual co-quantale, Girard co-quantale, pre-dual co-quantale and pre-girard co-quantale are discussed, and at the same time, some properties and equivalent characterizations of them are given respectively. It is proved that every co-quantale can be embedded into a pre-girard co-quantale. It is also proved that conuclei and ideal nuclei on Girard co-quantale are in one-to-one correspondence.Chapter Three:Fuzzy co-quantales. Firstly, we introduce the concept of fuzzy co-quantales by means of fuzzy Galois connections and give some examples of fuzzy co-quantales. The quantic nucleus and conucleus on fuzzy co-quantales are studied. Secondly, the concepts of fuzzy pre-dual co-quantale and fuzzy girard co-quantale are introduced. It is proved that L-quantic conucleus and L-ideal nucleus are one-to-one corresponding. Finally, the related categorical properties of the fuzzy co-quantales are discussed and the structures of the product, the limit and the inverse limit of the category of fuzzy co-quantales.Chapter Four:Fuzzy C-continuous lattices. Firstly, we introduce the concept of fuzzy nonempty Scott-closed set of fuzzy complete lattice and give the concept and properties of fuzzy beneath relation. Based of this, we give the notions of fuzzy C-continuous lattices and fuzzy C-algebraic lattices. It is proved that fuzzy complete lattice X is fuzzy C-continuous lattice if and only if ((?),∪) is a fuzzy Galois connection between X and FC(X).