Research On Fuzzy Quantale Congruence And Fuzzy Quantale

Aimed to set up new mathematical models for quantum mechanics, Quantale theory was produced in the1980s. Since D.N.Yetter found the interrelation between linear logic presented by Girard and the theory of quantales in1990, the theory of quantales has aroused great interests of many scholars and experts, especially a lot of academicians from the areas of mathematics and theoretical computer science. The new research and application of the quantale theory have developed a lot in recent thirty years. Since2000, the fuzzy set theory has been applied to Domain theory, forming fuzzy Domain theory. So, it’s not difficult to think of applying the thought of fuzzy poset theory to Quantale theory. In2012, Dr Wang Kaiyun introduced the concept of fuzzy quantale, discussed the properties of fuzzy nuclei and fuzzy conuclei, and given the definition of fuzzy quantic quotient and its characterization, etc. In this thesis, the concepts of fuzzy quantale congruence and quotient fuzzy quantale are introduced, the relationships between the fuzzy quantale congruences and fuzzy nuclei on a fuzzy quantale, the special fuzzy nuclei and fuzzy quantic quotients are discussed. At last, it is proved that three categories of special fuzzy quantales are reflective full subcategories of the fuzzy quantale category.The structure of this thesis is organized as follows:Chapter1:Preliminaries. The basic concepts and results on fuzzy posets, fuzzy quantales and category theory which will be used throughout the thesis are given.Chapter2:Fuzzy quantale congruence. Firstly, the concept of fuzzy quantale congruence is introduced and its properties are discussed. Secondly, it is proved that there is a one to one correspondence between the fuzzy quantale congruences and the fuzzy nuclei on a fuzzy quantale. At last, the concept of quotient fuzzy quantale is given and relevant isomorphism theorem is obtained.Chapter3:Fuzzy quantic quotient. Firstly, the basic properties of fuzzy quan-tale nuclei are discussed, and the set of fuzzy quantale nuclei is proved to be a complete lattice. Secondly, the related properties of three fuzzy quantic quotients are studied, the largest commutative fuzzy quantic quotient (right-sided quantic quotient、two-sided quantic quotient)is obtained. Finally, it is proved that the cat-egory of all commutative fuzzy quantales(right-sided fuzzy quantales and two-sided fuzzy quantales)and fuzzy quantale homomorphisms is a reflective full subcategory of the category of all fuzzy quantales and fuzzy quantale homomorphisms.