A Number Of Issues. The Domain Theory

This thesis investigates several current problems in domain theory. The thesis divided into two parts. In the first part, a correspondence between normal subsets of a base and Continuous sub-domains in a continuous domain will be established. Then an algebraic characteristic of Continuous sub-domains that is independent of embedding- projection pairs will be given based on the correspondence. This result is important for applications to develop an information system-based method for solving effectively continuous domain equations. In the second part, we proposed an ideal framework for quantitative domain theoiy and obtained some interesting results partly based on the ideas of stratifications in L-Fuzzy theory. One can investigates various topics, such as special domains, Scott topology and power domains and so on, similar i the 搤ssical domain theory and metric theory. Meanwhile it provides ~ 慽nk between quantitative domain theory. L-Fuzzy theory and presheaf models of concurrency. Moreover, the methods used in the thesis depended only on the adjoint properties of a frame, so can be applied more general framework such as the theories of ca-categories of Wagner and enriched category of Lawvere. In chapter 2 of the thesis, we established a correspondence between normal subsets of a base and continuous sub-domains in a continuous domain, gave an algebraic characteristic of continuous sub-domains that is independent of embedding -projection pairs based on the correspondence and proved a fixed point theorem on continuous mappings on the class of abstract bases. From these results, an information Istem ?ised method for solving effectively continuous domain equations c攗ld be obtained. As in the case of algebraic domains, one could solve equations with equalities at the base level using the method. In chapter 3 and 4 of the thesis, we investigate the quantitative domain theory through the viewpoints of L-Fuzzy theory. In chapter 3 we proposed a category of L-Fuzzy preordered sets and L-Fuzzy monotone mappings, proved a representation theorem about L-Fu.zzy preordered sets, that is, every L-Fuzzy preordered set VI / equivalent to a sheaf of preorders. Then we established a satisfied adjoint theory on L-Fuzzy monotone mappings that generalizes simultaneity the theories of ordinary and Rutten抯 ultrametric adjoints. In the final chapter 4, we set up a new theory of stratified convergences in the category of L-Fuzzy preordered sets based a simple and intuitive idea, that is, the sequence (1/ti) often could replaced the [0, 1] in computing practices. We then explored the relative theory about completions, fixed point theorems and quantitative domain equations by using the stratified theory. Some papers have worked out based on results of the thesis, that is, [4, 5, 6, 9 ? 12] in Bibliography. Amongst of these papers, [4] have published in 揊uzzy Systems and Mathematics?(in Chinese), [5] will appear in 揂dvances in Mathematics?(in Chinese), (6] have accepted by 揚roceedings of First International Confered~ on Domains, Shanghai, China 1 999?Domain and Process, eds by K.Keimel, G.Q.Zhang), and [9] have accepted by 揚roceedings of Mathematical Foundations of Program Semantics XVII, Aarhus, Denmark 2001? [10]. [11] and [12] have been submitted as we...