Exact Domain And FZC - Continuous Posets

Domain theory prepared the mathematic ground for semantics of programming languages. It is characterized by the close connection between order and topology, which makes it the common study field of both computer experts and mathemati-cians. And the generalization of domain is an important content in the theoretical of domain. So far, the more successful generalization are quasicontinuous domain and Z-continuous poset. As an another extention of Domain, Mashbum introduced the concepts of Weakly way below relation、Exact poset and weak domain in 2007, and discussed some basic properties of them. Since the concept of subset system was in-troduced 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. Liu Min and Zhao Bin introduced a new kind of continuity——FZ-continuity. Based on it, the concept of FZ-Domain is presented. In this paper, we introduced W-algebraic poset, FZC-continuous poset. A series of properties of them relating to order and topology are studied.The arrangement of this thesis is as follows:Chapter One:Preliminaries. In this chapter, the basic concepts and existing results of the theories of lattice, domain, topology and Z-subset system which will be used throughout the thesis are given.Chapter Two:Exact domain. Firstly, some properties of Weakly way below relation and the condition that the fixed points of Scott continuous is a Exact Domain are discussed. Secondly, the concepts of W-algebraic posets and strong W-algebraic poset are introduced. The image of a W-algebraic poset under a special operator has to be W-algebraic poset. Thirdly, it is showed that Exact domains or W-algebraic domains are hereditary for Scott open subsets and Scott closed subsets. Finally, the definition of local basis is introduced. We obtained some equivalent characterizations of local basis of weak domain.Chapter Three:FZC-continuous poset. The definitions of FZ-Lawson topology and Bi-FZ-Scott topology are given. Some fundamental properties about FZ-Lawson topology and Bi-FZ-Scott topology are obtained. Simultaneously, the concept of FZC-continuous poset is introduced and an equivalent characterization of FZC-continuous poset is given by using Galois Connections.