Research On Some Questions In Quantic Lattices And Quantales

Quantales were initially introduced by C. J. Mulvey in 1986, by replacing the intersection operation in the lattice of open sets of a topological space with a non-commutative binary operation &, with the intent of offering a new mathematical model for Quantum Mechanics when studying the spectrum theories of non-commutative C^*-algebras. In 1990, D. Yetter found the interrelation between linear logic, the logical foundation of theoretical computer science, which was presented by Girard, and the theory of quantales. Since then, the theory of quantales has aroused great interests of many scholars and experts, and a great deal of new ideas and applications of quantales have been proposed. The contribution of this thesis to the theory of quantales can be divided into two parts. The first one is to apply the methods of frames to quantales and to investigate some algebraic properties and some topological characteristics of quantales on the basis of the existing achievements, which enrich the theory of quantales. The second part is devoted to discuss the interior structures and categorical properties of quantic lattice for the categorical point of view. This part generalized, to some extent, the theory of quantales. The structure of this thesis is organized as follows:Chapter One: Preliminaries. In this chapter, we recapitulate the basic concepts and existing results of the theories of lattices, quantales and categories which will be used throughout the thesis.Chapter Two: Filters in quantales. In this chapter, the definition of filters in two-sided quantales is firstly given. We present some characteristics of filters in quantales, study the topological characteristics of filters space in quantales, and obtain a sufficient and necessary condition for filters space to satisfy the T₀ separation axiom. Secondly, the definition of prime filters is given. We discuss the topological properties of dual prime spectrum space of quantales, and prove a series of interesting conclusions. At last, the concept of fuzzy filters on quantales is given. We study the relationship between filters and fuzzy filters, and find the interrelation between morphisms of quantales and fuzzy filters.Chapter Three: The connection properties of quantales. The definitions of connection and local connection in frame theory are generalized to right-side idempotent quantales. We study a series of properties of connection and local connection, and obtain a sufficient and necessary condition for a quantale to be connected. We prove that some classical theorems in the theory of frames are still valid in quantales.Chapter Four: Girard quantale. Firstly, the interior structure of Girard quantalcs is studied, the relation of the operations in this structure is discussed, and thereafter some equivalent characteristics of Girard quantales are given. Secondly, the cyclic dualizing elements in Girard quantales are researched systematically. An example saying that in Girard quantales the cyclic dualizing element is not unique is given. A sufficient condition for the uniqueness of cyclic dualizing element in Girard quantales is obtained. At last, we prove that there is a one-to-one correspondence between the set of all the cyclic dualizing elements in Girard quantales and the set of all the unary operations satisfying a certain condition.Chapter Five: The category of quantic lattices. Quantic lattices can be seen as a generalization of quantales and prequantales. They play an important role in quantum logics. In this chapter, we firstly discuss the interior structure of quantic lattices, study some properties of sub-quantic lattices and quotient quantic lattices, and give the definition of the congruence relations of quantic lattices and the definition of quantic lattice nucleus. We prove that in a quantic lattice there is a one-to-one correspondence between the set of all the quantic lattice nuclei, the set of all the congruence relations, and the set of all the quotient objects. Secondly, We study the properties of quantic lattices from the categorical point of view. We prove that the category of quantic lattices has equalizers and co-equalizers, find its limit structure, and hence prove that it is a complete category.