基于 quantaloid 的多值收敛空间理论

The theory of many-valued topology is an interdisciplinary of general topology and many-valued logic. Höhle, a famous German scholar, is the founder of this theory. In 2001, he developed a theory of many-valued topology based on quantale, and this theory includes almost all of the familiar theories of fuzzy topologies. In 2014, he introduced a theory of many-valued topology based on quantaloid, and this theory gives a common framework for the theory of many valued topology based on quantale as well as for the theory of non-commutative topology. In addition, the applicant find that this theory includes a new theory of fuzzy topology which is different from the previous theories of fuzzy topologies. This makes the theory of many-valued topology based on quantaloid has more borad research value. In the theory of (many-valued) topology, the theory of convergence is one of the core contents, and many concepts and properties of topology can be described by it. The aim of this project is to establish the convergence theory corresponding to the theory of many-valued topology based on quantaloid, the contents include the basic notions, properties and the topological conditions of quantaloid-valued convergence space. The expected results have specifically theoretical significance to enrich and develop the theory of many-valued topological space.

多值拓扑理论隶属一般拓扑学与多值逻辑的交叉领域.德国知名学者 Höhle 是这一理论的奠基人.2001年,他引入了基于 quantale 的多值拓扑理论,该理论几乎涵盖了常见的模糊拓扑理论.2014年,他又引入了基于 quantaloid 的多值拓扑理论,该理论又把基于 quantale 的多值拓扑理论与非交换拓扑理论纳入到统一框架下.另外,申请人发现该理论涵盖了一种不同于已往模糊拓扑的新模糊拓扑理论.这使基于 quantaloid 的多值拓扑理论具有更广泛的研究价值.收敛理论是(多值)拓扑学的核心内容之一,很多概念和拓扑性质都可通过收敛来描述.本项目旨在建立与基于 quantaloid 的多值拓扑理论相应的收敛理论,主要内容包括 quantaloid 值收敛空间的基本概念、性质及拓扑化的条件,其预期成果对于丰富和完善多值拓扑空间理论体系具有鲜明的理论意义.

格值收敛空间是格值拓扑空间的自然推广，本项目围绕格值收敛空间理论做了如下研究。首先，系统研究了几种格值收敛空间之间的范畴关系；其次，讨论了格值收敛空间的拓扑化条件、正则性和连通性；再次，建立了基于格值邻域系空间(等价于预拓扑的格值收敛空间)的格值邻域群理论；最后，借鉴格值拓扑（收敛）的研究方法，探讨了格值凸结构的范畴性质，以及格值粗糙集的公理化问题。在本项目中，我们一方面发展和丰富了格值拓扑理论的基本内涵；另一方面还将相关方法应用于格值凸结构和格值粗糙集的研究。我们的研究工作不仅有重要的理论意义，而且还有潜在的应用价值----基于模糊邻域系的格值粗糙集，提供了粗糙集理论与应用的新思路和新方法。