加载群结构的格值收敛理论的研究

Algebra, topology and order are closely related to each other. The notion of lattice-valued topological groups is a theory of group-based lattice-valued topological spaces, and it combines the three structures of group, topology and order together. In 2014, Ahsanullah etal., established a theory of lattice-valued convergence group by replacing lattice-valued topology with more general lattice-valued convergence structure.Due to the short time, the research of this theory is still in the start stage and there are many problems to be solved. This project aims at the improvement and development of the theory of lattice-valued convergence group, and it will be carried out from the following three aspects: (1)The theory of lattice-valued convergence group based on lattice-valued filter and lattice-valued net. We will research its homogeneous property, localization, uniformizability, completion and topologicalness, etc. (2)The categoric properties of lattice-valued convergence group and its relationships with related structures. (3)Furthermore, we will fuzzify the group structures, and then study the theory of lattice-valued convergence group based on fuzzy groups.The accomplishment of this project will increase our understanding of the connection between different mathematical structures, through which the theory of many-valued topology will be further enriched and improved, making contributions to the mathematical theory of fuzzy sets.

代数、拓扑和序结构有着紧密而自然的联系。格值拓扑群是加载群结构的格值拓扑理论，它将群、拓扑和序三种结构有机的结合在一起。2014年，把格值拓扑替换成更一般的格值收敛结构，Ahsanullah等建立了格值收敛群理论。因时日尚短，该理论的研究尚处于起步阶段，还有不少问题亟待解决。本项目旨在完善和发展这一理论，拟从以下三个方面展开研究：(1)基于格值网和格值滤子的格值收敛群理论，研究其齐性、局部化、一致化、完备化和拓扑化等问题。(2)格值收敛群的范畴性质及其与相关结构的范畴关系。(3)进一步将群结构模糊化，研究模糊群上的格值收敛群理论。本项目的完成将深化我们对不同数学结构之间联系的认识，丰富和完善格值拓扑的理论体系，为模糊集的数学理论的发展做出贡献。

在模糊集统一背景下，本项目研究了格值拓扑（收敛）和格值粗糙集理论。首先，通过对角条件和邻域系两种方式探讨了格值收敛空间的拓扑化问题；其次，研究了格值收敛空间的正则性，特别是建立了其连续函数的扩张定理；再次，引入了新的格值拓扑群结构，证明该结构具有很好的范畴性质，并与相关结构具有相容的范畴关系；最后，借鉴格值拓扑的研究方法，构建了格值粗糙集的新模型，并研究了其公理化问题。相关成果丰富和发展了格值拓扑理论的基本内涵；提供了粗糙集理论与应用的新思路和新方法。