Pseudo Continuous Co-quantale And Fuzzy Pre-dual Quantale

The concept of quantale was introduced in1986by C.J.Mulvey to study noncommutative C*-algebra and provide a new mathematical model for quantum mechanics. In recent twenty years, the theory of quantale has been paid attention by mathematicians and logicists, lots of new ideas and applications of quantale have been established. The first part of this thesis is to further study classic quantale theory, the concept of pseudo continuous co-quantale is introduced, some properties, ideas and categorical properties of the pseudo continuous co-quantale are studied carefully and deeply. The second part of this thesis is to define the concepts of fuzzy pre-dual quantale and fuzzy pre-girard quantale on the base of fuzzy quantale and study the L-(pre-)nuclei and L-(pre-)ideal conuclei of them. At the same time, the concept of homomorphism of fuzzy dual quantale is introduced, some categorical properties of the fuzzy dual quantale are studied. The structure of this thesis is organized as follows:Chapter One:Preliminaries. In this chapter, we give some basic concepts and results of the quantale theory, co-quantale theory, category theory and the fuzzy quantale theory which will be used throughout the thesis.Chapter Two:Pseudo continuous co-quantale. Firstly, the concepts of pseudo continuous co-quantale and pseudo semi-continuous co-quantale are given, some properties of pseudo continuous co-quantale and pseudo semi-continuous co-quantale are studied. Secondly, the definition of ideal on the pseudo continuous co-quantale is introduced, the properties of ideal on the pseudo continuous co-quantale are stud-ied. Finally, some categorical properties of the pseudo continuous co-quantale are obtained.Chapter Three:Fuzzy pre-dual quantale. Firstly, the definitions of L-pre-nuclei and L-pre-ideal conuclei are given, some properties of them are studied. Secondly, the concepts of fuzzy pre-dual quantale and fuzzy pre-girard quantale are introduced, the relationship between fuzzy pre-dual quantale and fuzzy pre-girard quantale is discussed. Finally, the relationship between L-(pre-)nuclei and L-(pre-)ideal conuclei on the fuzzy pre-dual quantale are discussed, it is proved that there is a one-to-one correspondence between L-(pre-)nuclei and L-(pre-)ideal conuclei on the fuzzy pre-girard quantale.Chapter Four:The category of fuzzy dual quantale. Firstly, the concept of homomorphism of fuzzy dual quantale and the structure of equalizer of fuzzy dual quantale are given. Then, we prove that the category of fuzzy dual quantale has the product. Secondly, the limit on the category of fuzzy dual quantale is constructed. Finally, the concept of the inverse system on the category of fuzzy dual quantale is introduced, the inverse limit of the inverse system on the category of fuzzy dual quantale is constructed.