相容幂domain结构与函数逼近结构相关问题研究

Domain theory is an important field generated by the intersection of General Topology and Theoretical Computer Science. Powerdomain theory is one of very important subfields of Domain theory. The most important topic of the theory is about how to effectively construct the structure of powerdomains, and it is also a very difficult topic to be solved. The project focuses on three consistent semilattice operators, and considers the existence and construction of free algebras over a continuous domain, and tries to characterizing three powerdomains by applying the techniques of Function space and FS-domain in a totally new way. The fundamental Approaches applied by the research can be put in three points. One is to construct the three consistent powerdomains and their topological propositions, another is to define three FS-domains for three partial operators, that is to classify three consistent powerdomains by the subcategories of FS-domain category, and the other is to prove three consistent powerdomains on FS-domains are FS-domains. The project is trying to apply more techniques of general topology to characterizing the three powerdomain constructions and the functional approximations. For advancing the integration among the Order theory, Topology and Domain theory, and for providing powerful mathematics background to solve the remain problem of theoretical computer science.

Domain理论是一般拓扑学与理论计算机科学相互交叉而生的重要领域。幂domain理论是Domain理论的重要分支。如何有效地构造幂domain结构，一直是该理论最主要的问题，也是研究难度很大的问题。本项目研究三类相容幂domain结构与三类函数逼近结构相关问题。研究思路有三：其一，构造三类相容幂domain结构，并完善其性质；其二，利用函数逼近思想，实现三类函数逼近结构对三类相容幂domain结构的函数式刻画；其三，建立FS-domain范畴关于相容幂domain结构的封闭性。本项目期望通过引入一般拓扑学的诸多技巧研究三类相容幂domain结构及其与三类函数逼近结构之间的密切关系，进而推进一般拓扑学、格序理论，以及Domain理论的交叉结合，并为理论计算机领域中的一些遗留问题（如FS-domain与双有限domain的收缩之间的等价性问题）提供更有力的数学理论支持。

Domain理论是一般拓扑学与理论计算机科学交叉而成的较新研究领域。本项目主要研究的幂domain结构理论，是该交叉领域的重要分支。项目研究的主要成果包括：1.利用相容交算子引入相容交FS-domain，并证明以相容交FS-domain为对象，以Scott连续函数为态射的范畴是Cartesian闭范畴，采用相容交FS-domain刻画Lawson紧连续domain上的相容上幂domain。2.利用相容并算子引入相容线性FS-domain（即相容并FS-domain），证明了以相容并FS-domain为对象、以Scott连续函数为态射的范畴是Cartesian闭范畴。3.有界完备domain是完全分配的当且仅当它是分配的相容并FS-domain；带有最小元的dcpo L是连续的当且仅当L上的相容下幂domain是连续的当且仅当L上的相容下幂domain是分配的相容并FS-domain。4.FS-domain上的三类相容幂domain仍是FS-domain，即三类相容幂domain结构对FS-domain范畴是封闭的。上述成果的部分内容已被撰写成4篇学术论文，其中1篇发表在理论计算机学科重要期刊《Electronic Notes in Theoretical Computer Science》上、 1篇发表在重要计算机期刊《计算机科学》上，另外2篇已投稿在国际一般拓扑学核心期刊《Topology and its Applications》。