S*-双连续偏序集的B-拓扑性质、拓扑表示及笛卡尔闭性质研究

Domain Theory is a very important research topic which originates from Theoretical Computer Sciences and Mathematics. The key characteristic of Domain Theory is the fusion of order and topology structure. A significant research content of Domian Theory and Order-Topology Theory is the conyinuity of posets and the related topologies. Based on the existing results, the project is going to work on the following four aspects: (1) Using the structure theorem for B-topology, exploring the sufficient and necessary conditions for the O1 and q-convergences being topological; (2) Providing the topological representation for S*-double continuity by studing the locally compact Properties of B-topology and the structural feature of morphism set consisting of all double Scott continuous maps; (3) Developing the equivalent characterization for the Cartesian Closed Properties of S*-double continuity, and exploring Cartesian Closed Subcategories of S*-doubly continuous posets; (4) By studing the relationships among some kinds of continuity of posets, searching the order-theoretical conditions under which the B-topology agree with the double Scott topology、the interval topology and the Lawson topology. Also we look for the order-theoretical conditions such that the B-topology is inherited and product consistency.

Domain理论是当前理论计算机科学与数学理论研究的重要课题，序结构与拓扑结构的交叉融合是该理论的研究特点。序结构的连续性与相关拓扑研究是Domain理论与序拓扑理论研究的重要方向。本项目拟结合现有成果进行以下四个方面的研究：(1) 通过偏序集上B-拓扑的结构表现定理研究O1-收敛与q-收敛的可拓扑化问题，得到O1-收敛与q-收敛可拓扑化的等价刻画条件；(2) 通过研究S*-双连续偏序结构上B-拓扑的局部紧性质以及双Scott连续态射空间的基本结构特征得到S*-双连续集的拓扑表示；(3) 实现S*-双连续偏序集范畴笛卡尔闭性质的等价刻画，寻求该范畴的笛卡尔闭的满子范畴；(4) 借助于偏序结构上各个连续性之间的关系，寻找B-拓扑与双Scott拓扑、区间拓扑保持一致的条件，以及B-拓扑具有遗传一致性与乘积一致性的条件。

20世纪40年代，由著名数学家Birkhoff在偏序集上所引入的B-拓扑是联系序结构与拓扑结构的重要拓扑. 本项目研究以S*-拟双连续偏序集上的B-拓扑为主要研究对象，得到了若干结果，对序收敛理论与序拓扑理论的完善与发展具有一定意义，具体地，..(1) 在偏序集上引入OS-序收敛并证明了OS-收敛所生成的拓扑为B-拓扑，深度揭示了序收敛与B-拓扑的内在关联；..(2) 给出了S*-拟双连续偏序集上B-拓扑的基本性质，在范畴层面实现了S*-拟双连续性的B-拓扑表示，丰富了B-拓扑的研究结果；..(3) 得到了B-拓扑与双Scott拓扑(Lawson拓扑、区间拓扑)一致的充分与必要条件，解决了B-拓扑具备Dedekind-MacNeille完备遗传保持性的等价刻画，为序拓扑理论中基本问题的解决提供了新的思路与方法.