基于quantaloid-加载范畴的quantale值收敛理论

项目名称： 基于quantaloid-加载范畴的quantale值收敛理论

项目编号： No.11471297

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 方进明

作者单位： 中国海洋大学

项目金额： 62万元

中文摘要： 格值收敛空间因较格值拓扑空间更具一般意义而成为格上拓扑学的研究对象。近年来，模糊集的范畴论基础的最直接结果是任意quantale可诱导quantaloid、任意的quantaloid对应的quantaloid范畴可解释成具有多值序结构的模糊集。结合最近我们格值收敛理论的科研工作，我们团队认识到用quantaloid范畴的理论和方法，在模糊集上而不是在经典集合上建立quantale值收敛理论是现有格值收敛理论的努力方向。本项目的目的是：（1）建立模糊集上的quantale值收敛理论，其包括quantale值广义收敛理论和quantale值拟一致收敛理论两个部分；（2）建立格值拓扑空间、quantale值广义收敛空间、quantale值拟一致收敛空间的内在联系，并指明quantale值拟一致收敛结构是较格值拓扑更具一般意义的格值空间结构。这些研究结果将为一般模糊集上格值收敛理论的展开奠定基础。

中文关键词： 模糊集；模糊拓扑；Quantale理论；Quantaloid-加载范畴；格值收敛

英文摘要： Lattice-valued convergence space plays a more important role than lattice-valued topological space in topology on lattice since it has more general sense than lattice-valued topological space. Recently, the researching results on the categorical foundation of fuzzy sets confirm that from a given quantale, a quantaloid can be constructed, and for a small quantaloid, a corresponding quantaloid category can be interpreted as a fuzzy set endowed with a many-valued order. Combining with the researching works of our group about lattice-valued convergences, we realize that the striving direction in the theory of lattice-valued convergences is to establish a theory of quantale-valued convergences on a fuzzy set instead of a crisp set in the way of using the theory of quantaloid-enriched category. The purpose of our program is twofold: (1) we plan to establish a theory of quantale-valued convergences, which is composed of quantale-valued generalized convergences and quantale-valued quasiuniform convergences; (2) From the viewpoint of the theory of category, the relationship among latice-valued topological spaces, quantale-valued generalized convergence spaces and quantale-valued quasiuniform convergence spaces will be established, and then we hope to confirm that quantale-valued quasiuniform convergence structures introduced in our program have more general sense than lattice-valued topologies. These results obtained in our program will offer a foundation for the development of lattice-valued convergences on a fuzzy set.

英文关键词： Fuzzy set;Fuzzy topology;The theory of quantale;Quantaloid-enriched category;Lattice-valued convergence