基于lax代数的多值拓扑理论

Topology and algebra are closely related to each other. Since 2001, a group of scholars led by Tholen、Clementino、Hofmann、Seal established the theory of lax algebras, which provides a general framework for several important structures in classical topology such as metric spaces, closure spaces and topological spaces. This project aims at a systematic investigation of the theory of many-valued topology based on lax algebras, incorporating existing theories of many-valued topology on crisp sets into the framework of lax algebras, as well as establishing a fairly complete theory of many-valued topology on fuzzy sets through lax algebras. For the latter, in particular, due to the lack of understanding of fuzzy powersets of fuzzy sets, there are no satisfactory results among the existing attempts of creating topological structures on fuzzy sets; to solve this problem, the theory of lax algebras will provide a useful toolkit. The accomplishment of this project will increase our understanding of algebraic structures of many-valued topology, through which the theory of many-valued topology will be further enriched and improved, making contributions to the mathematical theory of fuzzy sets.

拓扑与代数有着紧密而深刻的联系。从2001年起，以Tholen、Clementino、Hofmann、Seal为首的一批学者建立了lax代数理论，为度量空间、闭包空间、拓扑空间等经典拓扑学中的重要结构提供了统一的理论框架。本项目旨在系统研究基于lax代数的多值拓扑理论，一方面将分明集上的现有多值拓扑理论纳入lax代数理论的框架下，另一方面利用lax代数在模糊集上建立较为完善的多值拓扑理论。特别是后者，由于人们对模糊集的模糊幂集认识不足，在模糊集上建立拓扑结构的尝试一直没有取得满意的结果，而lax代数理论将为该问题的解决提供有效的工具。本项目的完成将深化我们对多值拓扑的代数结构的认识，进一步丰富和完善多值拓扑的理论架构，为模糊集的数学理论的发展做出贡献。

本项目从lax代数的角度研究多值拓扑和多值序结构。所得结果反映了范畴论在模糊集理论研究中的重要应用。特别地，lax代数理论在模糊集的背景下将“拓扑”与“代数”有机结合在一起，这也是本项目的主要特色。有代表性的成果列举如下：（a）在Q-范畴的背景下揭示了正则性与完全分配性的联系。（b）建立了模糊集上的模糊序理论。（c）建立了多值非对称性理论。（d）给出了拓扑范畴中的稠密性与Q-范畴中的稠密性的联系。（e）证明Q-分配子与对角构成的范畴等价于Q-内部空间与连续关系所构成的范畴的商范畴。