Research On Quantales And Four Types Of Quantale Structures

Quantale was proposed by C.J.Mulvey in 1986, with the intent of offering a new lattice-characterization for studying non-commutation C*-algebras and a new mathematicial model for quantum mechanics. The term quantale was coined as a combination of "quantum logic" and "locale" by C.J.Mulvey. In 1990, D.Yetter found the interrelation between linear logic, the logical foundation of theo-retical computer science, wihch was presented by Girard, and the theory of quantales. Since quantale theory provides a powerful tool in studying noncommutative struc-tures, it has a wide applications, especially in studying noncommutative C*-algebra theory, the ideal theory of commutative ring, linear logic and so on. The quantale theory has aroused great interests of many scholar and experts, a great deal of new ideas and applications of quantale have been proposed in twenty years. The first Part of this thesis is to apply the methods of frames theory and category theory to in-vestigate algebraic properties, topological charateristics and categorical properties of quantales on the basis of the existing achievements, which enrich the theory of quan-tale. The second part is to study the interior charateristics and categorical properties of the related structures of quantale, with the intent of offering new growing point for quantale and generalized the theory of quantale. The strucure of this thesis is organized as follows:Chapter One:Preliminaries. In this chapter, we gave the basic concepts and existing results of the theories of the theories of lattices, quantale and categories which will be used throughout the thesis.Chapter Two:Diameter and Hausdorff property on Quantales. Firstly, the definition of point in quantale is given. We discussed some properies of point, and the equivalent characterization for T-spatial quantales with point was given. Secondly, the definition of diameter on frame is generalized to quantales. We study some porperies of the topology induced by the diameter of quantale, some charaterization of the diameter of quantale are given. Thirdly, the two concepts of convergence on quantale is introducted, the definition of the cluster point, limit point, strong limit point and completeness of quantale are given. We study some properties of the convergence on quantale, and some interesting conclusions were obatined. Finally, the definition of T2,T2* and T2** of quantale are given, We discussed some relations of them, and some equivalent charaterizations were obtained.Chapter Three:Firstly, the concept of saturated element in quantale is intro-duced, the structure of the coproduct of the unital quantales is obatined. Secondly, the definition of bimorphism of quantale is given, the tensor product of quantales is investigated and their properties are discussed. Finally, the definition of weakly spatial locale, completely regular locale and zoro-dimensonal locale are generalized to quantales. Their properties are discussed and prove that some classical theorems in the theory of locales are still valid in quantales. Some conreflection subcategories of the subcategories of quantales are obtained.Chapter Four:The category of double quantale modules. Firstly, the definition of bimorphism of double quantale modules is given, the tensor product of double quantale modules is obatined, and some properties of their is discussed. Seconedly, base on the limit of the category double quantale modules, the dirct limit of the category double quantale modules is obtained. Finally, We present some interesting examples of double quantale modules, the free objects of the double quantale modules is obatined. It is prove that the category of double quantale modules is algebric.Chapter Five:The Q-fuzzy order and E-quantales. Firstly, the definition of Q-fuzzy order and Q-fuzzy equivalence are given, we study the relation of Q-fuzzy order and cover. It is prove that the class of all Q-fuzzy order relations on the non-empty set is a quantale, and the cut set of Q-fuzzy equivalence exactly as equivalence when the range is a frame. Secondly, the definition of E-quantale is given. An embedding functor from the of category of quantales to the category of E-quantale is introducted. The properties ideal and covers of E-quantale are discussed. Finally, the definition of admissible set and generalized Boolean algebra are given. Their properties are discussed and some characterization of them are obtained.Chapter Six:The category of L-fuzzy quantales. Firstly, the definition of L-fuzzy subframe is generalized to L-fuzzy quantale, and some equivalent charater-ization of L-fuzzy quantale are given. Secondly, the concepts of L-fuzzy ideal, and L-fuzzy prime ideal are introduced, their properties are discussed. Finally, we study the properties of L-fuzzy quantales from the categorical point of view, the product, equalizers, co-equalizers, meet and pullback of the category of L-fuzzy quantales are investigated, and the structure of the limit of the category of L-fuzzy quantales is obtained. Hence the category of L-fuzzy quantales is completed.