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。

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.