关于spectral集和spectral拓扑若干问题研究

In this project, from the point of view of the intersection of ordered structure, topology and algebra, we will investigate systematically the stably compact representations of posets and the spectral topologies and Priestley topologies on representable posets, and hope to solve some relative open problems. The hypercontinuity of the lattice of all ideals of a distributive lattice will be discussed and the topological characterizations based on the Stone topology and Priestley topology on the prime filters of a distributive lattice will be given. The order properties such as complete distributivity, continuity and hypercontinuity of the lattice of clopen upper sets in Priestley spaces will be investigated. In this way, the dualities established by Stone, Priestley and Easkia would be devloped. Some ordre theoretical charaketerizations of the sobrity of Scott topology on a dcpo will be obtained. By these ways, we will provide some important connections among the order, topology and algebra.

从序、拓扑、代数的交叉角度，系统地研究偏序集的稳定紧可表示问题和spectral可表示偏序集上的spectral拓扑和Priestley拓扑，解决相关的遗留问题；研究分配格之理想格的超连续性，基于分配格素谱的Stone及Priestley拓扑，给出相应的拓扑刻画；刻画Priestley 空间中由上开闭集格（赋予集包含序）的完全分配性、连续性和超连续性等序性质，并给出相应范畴的对偶等价定理，发展Stone、Priestley和Easkia所建立的对偶理论；研究dcpo上Scott拓扑的sober性, 获得序刻画。基于这些研究，为序、拓扑、代数之间的交叉提供若干新的联结。

本项目从序与拓扑的交叉角度, 深入研究了Domain理论和non-Hausdorff拓扑中的一些基本问题和基本结构, 解决了多个涉及sober空间、well-filtered空间、d-空间和co-sober空间长久遗留的公开问题; 引入和研究了三类新空间Rudin空间、well-filtered决定的空间和强d-空间; 对一般的K-反射问题, 给出了一个统一的直接处理方法, 证明了K-反射函子具保拓扑空间的任意乘积; 建立了一个讨论sober空间、well-filtered空间、d-空间及相关空间的统一框架, 发展了一个统一的理论和处理方法; 发展了若干研究Domain理论和non-Hausdorff拓扑的新思路、新方法及技术; 为深化对sober空间、well-filtered空间、d-空间及相关空间及其结构的了解, 提出了30多个公开问题. 本项目的研究工作进一步促进了序与拓扑的交叉融合, 丰富了Domain理论和non-Hausdorff拓扑.