Domain理论中的收敛性与K-有界Sober空间范畴

Domain理论作为格上拓扑学与理论计算机科学中的前沿分支学科，有较大的理论研究价值和广泛的应用前景。本项目将在前期工作的基础上深入研究Domain理论中的收敛性与K-有界Sober空间范畴，拟进行以下三个专题研究：(1)关于Quantale值SI-余拓扑空间的研究，重点是解决Quantale值余拓扑空间中SI-收敛的可拓扑化问题；(2)关于偏序集中Liminf收敛的研究，重点是确定一类生成拓扑恰是Lawson拓扑的收敛，建立模糊偏序集中L-滤子的Liminf收敛性；(3)关于K-有界Sober空间范畴的研究，重点是研究K-有界Sober偏序集范畴中的乘积结构，寻找T0空间范畴(态射是保存在既约集并的连续映射)的极大满子范畴，使得K-有界Sober空间范畴是该范畴的反射子范畴。本项目获得的结果不仅能够丰富量化Domain理论，也将促进Domain理论的深化与拓展。

As an important branch of topology on lattices and theoretical computer science, domain theory has much theoretical study value and wide application prospect. On the basis of the previous work, the aim of this project is to deeply study the convergences and the category of K-bounded sober spaces in domain theory. The project is arranged as follows: (1) Study on quantale-valued SI-cotopological spaces, the key problem is to give a characterization for the SI-convergence being topological in quantale-valued cotopological spaces; (2) Study on the Liminf convergence in posets, we focus on determining a convergence in posets such that the topology generated by the convergence is exactly the Lawson topology, and establishing the Liminf convergence of L-filters in fuzzy posets;(3) Study on the category of K-bounded sober spaces, we focus on studying the product structures in the category of K-bounded sober posets, and finding a maximal full subcategory of the category of T0 spaces (the morphisms are continuous mappings preserving existing irreducible suprema) such that the category of K-bounded sober spaces is a reflective subcategory of this category. The results of this project can not only enrich quantitative domain theory, but also promote the deepening and expansion of domain theory.