The Research About Neibing Quantale And (?)-Girard Quantale

Abstract Since the concept of quantale was introduced by C. J. Mulvey in1986, many mathematicians and logicists have paid attention to the theory of quantale. In twenty years, a great deal of new ideas and applications of quantale have been proposed. The research of these subjects has related to several area such as non-commutative C*-algebra, the ideal theory of rings, linear logic and the-oretic computer science. Quantale was introduced, with the intent of offering a new lattice-characterization for studying non-commutative C*-algebra and a new mathematical model for quantum mechanics. It provides a powerful tool in study-ing non-commutative structures. For quantale can be regarded a generalization as Frame, it also has full interior strucures. The first one of this thesis is to study carefully and deeply some properties of the pre-nuclei and pre-ideal conuclei on quantales, NeiBing quantale and (?)-Girard Quantale. The second part is devoted to discuss the categorical properties of NeiBing quantale and (?)-Girard Quantale for the categorical point of view.The arrangement of this thesis is as follows:Chapter One:Preliminaries. In this chapter, we give the basic concepts and results of the theory of quantale, lattice theory and that of category theory which will be used throughout the thesis.Chapter Two:The pre-nuclei and pre-ideal conuclei on quantales. Firstly, some concepts, properties and conclusions of pre-nuclei on quantale are given. Secondly, the concepts, properties and conclusions of pre-ideal conuclei on quantale are introduced. There are characterizations of fuzzy ideals of quantale in terms of ordered fuzzy points. Thirdly, we give the relationship between pre-nuclear and pre-ideal conuclei on quantales.Chapter Three:NeiBing Quantale. Firstly, we introduce the concepts of Neib-ing element of quantale. give its related properties, and prove that the whole of quantale’s Neibing elements can construct of its sub-quantale. Furthermore, on the basis of Quantale’s Neibing element, we give the definition of NeiBing Quantale and its related properties were discussed. Finally, we discuss some related categorical properties of NeiBing quantale. Chapter Four:The categorical properties of (?)-Girard Quantale. Firstly, some concepts and related properties of (?)-Girard Quantale are given, and we also find out some examples of the (?)-Girard Quantale. Secondly, the concepts of sub-(?)-Girard Quantale is introduced, and some sub-(?)-Girard Quantale are gived, including the ideals of (?)-Girard Quantale. Thirdly, we discuss the related categorical properties of the (?)-Girard Quantale. We prove that the (?)-Girard Quantale category has equalizer, product, and it is completed.