Some Properties Of Quasicontinuous Domains And Order Homomorphisms On Continuous Local Dcpos

Since domain theory was introduced, a large number of new ideas and applications about domain theory have been given. As a generalization of con-tinuous domains and generalized continuous lattices, quasicontinuous domains were defined by G.Gierz, J.D.Lawson, and A.Stralka in 1983. And as a generalization of quasicontinuous domains and Z-continuous domains, quasiZ-continuous domains were proposed. Starting from a idea of the localized, continuous local directed complete sets were defined. These new arguments have greatly enriched domain theory, also aroused interests of many experts and scholars. In this paper, some properties of quasi-bases of quasicontinuous domains, character and density on qua-sicontinuous domains or quasiZ-continuous domains, and some properties of order homomorphism and extension theorem on local directed complete sets are discussed. The arrangement of this thesis is as follows:Chapter One Preliminary knowledge. In this chapter, we give the basic con-cepts and results of quasicontinuous domains, Lawson topology and quasiZ-contin-uous domains which are used in the paper.Chapter Two Quasi-bases of quasicontinuous domains and prequasicontinuous domains. Firstly, some properties of quasi-bases of quasicontinuous domains are discussed. Secondly, the relationship between quasi-bases and quasidirected minimal sets is studied, and the extension theorem on quasicontinuous domains are given. At last, we introduce the concept of prequasicontinuous domains, and discuss some properties about it.Chapter Three Character and density on quasicontinuous domains or quasiZ-continuous domains. Firstly, the concept of local quasi-bases on directed complete posets is given, and then character of quasicontinuous domains is defined. Some properties of local quasi-bases and relations between the character on a quasicontin-uous domain and those on the related spaces with Scott topology or Lawson topology are studied. Secondly, dense family is introduced, and density of quasicontinuous domains is given. It is proved that the density on a quasicontinuous Domain are respectively equal to those on the related space with Scott topology, but they are respectively less than or equal to those on the related space with Lawson topol-ogy. Finally, We promote quasi-bases, character and density to quasiZ-continuous domains, discuss and give the related properties and results.Chapter Four Order homomorphisms on continuous local directed complete sets. Firstly, the concept of local directed minimal sets is given, and then some properties about it are studied. Secondly, bases of continuous local directed com-plete sets is defined, and some equivalent characterizations of bases and the rela-tionship between bases and local directed minimal sets are discussed. Finally, order homomorphisms on continuous local directed complete sets is introduced, the related properties and the extension theorem of order homomorphisms are studied.