Researches On The Properties Of Quasialgebraic Domain And The Structures Of Other Domains

Since the concept of domain was first introduced by D.S.Scott in the 1970s, many mathemeticians and theoretical computer scientists pay close attention to the theory of domains. As the generalization of continuous domains and generalized continuous lattices, the concepts of quasicontinuous and quasialgebraic domain were defined by G.Gierz, J.D.Lawson and A.Stralka in 1983. The aim of this paper is to discuss some problems of continuous domains. On one hand, the properties of quasialgebraic domains are mainly studied. Then, we discuss another structure of domains-hypercontinuous domain. On the other hand, we also study deeply the properties of consistently continuous domain and connected algebraic domain. The arrangement of this paper is as follows:Chapter One Preliminary knowledge. In this chapter, we give some basic concepts and results of the theory of continuous domains and the theory of category which are used in the whole paper.Chapter Two Quasialgebraic domain and its relevant properties. In this chapter, we first study the properties of quasicontinuous domains by the mapping, and obtain the relationship among quasicontinuous domain , Scott continuous projective operator and Galois connection, respectively. And we give some relevant conclusions. Secondly, we give some basic properties of quasialgebraic domains by the mapping and topology, and get its equivalent propositions. Then, we also discuss deeply the topological cases. At last, we introduce the concept of hypercontinuous domain, give some equivalent discription about it, and show the relationship between hypercontinuous domain and continuous domain.Chapter Three Order — homomorphisms of consistently continuous domains. Firstly, we give the definition of consistently directed minimal set and argue about its some properties in this chapter. At the same time the relationship between consistently directed minimal sets and consistently continuous domains was obtained. Secondly, we give the concept of basis and its equivalent items. Besides introducing the order-homomorphism of consistently continuous domains, we study its some properties and obtain relevant extension theorems at last.Chapter Four Connected algebraic domain. In the final chapter, we study mainly consistent mapping of connected algebraic domain and give the structure of itsideals. According to these, we obtain that connected algebraic domain satisfies the ascending chain condition. At the same time we give that some equivalent propositions on consistently compact elements of connected algebraic domain and prove that the category of connected algebraic domains is equivalent to the category of posets.