Topology And Category Theoretic Aspects Of FZ-Domains

Abstract Domain theory prepared the mathematic ground for semantics of programming languages. It is characterized by the close connections between order and topology, which make it a study field of experts on both theoretical computer science and mathematics. Since the concept of subset system was introduced by Wright.Wagner and Thatcher in 1978, the study of posets in approach of Z-subset system has attracted widespread attention and many new ideas and approaches have been introduced. In this paper, a new kind of continuity——FZ-continuity is introduced in approach of Z-subset system and based on it the concepts of FZ-Domain and so on are defined. Some properties of FZ-Domain relating to order, topology and category are studied.The arrangement of this thesis is as follows:Chapter One:Preliminaries. In this chapter, the basic concepts and existing re-sults of the theories of domain, topology and category which will be used throughout the thesis are given.Chapter Two:FZ-Domains. Firstly, the FZ-way below relation is defined and based on it the concepts of FZ-continuous posets, FZ-Domain and so on are introduced. Some basic properties about them are studied. Secondly, the products of FZ-Domains and images of FZ-Domains under two kinds of projections are studied. Thirdly, the definition of algebraic FZ-Domain is given and some basic properties about it are studied. Finally, the definitions of FZ-basis and FZ-abstract basis are introduced. For one class of subset systems, we obtained some equivalent characterizations of FZ-Domains by directed FZ-basis and FZ-abstract basis.Chapter Three:The FZ-Scott topology. The definition of FZ-Scott open sets on Z-complete posets is given and, more over, it is proved that for one class of subset systems, the FZ-Scott open sets on a Z-complete poset form a topology, which is called FZ-Scott topology. Some fundamental properties about FZ-Scott topology are obtained. Some equivalent characterizations of the continuous maps for this topology in order theoretic terms are obtained, especially, for one class of subset systems, it is proved that a function f is a continuous map if and only if f preserves the supremums of directed Z sets. Finally, we focus on FZ-Scott topologies on FZ-Domains and for one class of subset systems, it is proved that they are Sober spaces if and only if they satisfy the Rudin property.Chapter Four:Two kinds of cartesian closed categories of FZ-Domains. Firstly, the definitions of the categories of FZ-Domain and FZCPO are given. Some ap-plications of them are given, for examples DOM, Poset, DCPO and some other important categories are specific examples of them. The category FZCPO is proved to be cartesian closed. Secondly, two kinds of subcategories of FZ-Domains——the category of finite separate FZ-Domains and the category of bifinite separate FZ-Domains——are proved to be cartesian closed, and some applications of them are given.