模糊拓扑结构和凸结构的松代数表示及应用

由模(monad)代数、关系代数和enriched范畴发展起来的松代数(lax algebra)理论是研究空间结构表示的重要方法，能有效地反映空间结构的代数属性。本项目的目的是当L是交换单位quantale时，利用松代数理论研究模糊拓扑结构和凸结构的刻画及其应用。首先，定义模糊半一致收敛空间，研究它的范畴性质并讨论正则完备性，基于模糊滤模和单位模分别给出模糊拓扑和模糊半一致收敛空间的松代数表示；其次，定义凸理想和凸理想模，研究凸空间的Kleisli幺半群刻画和松代数表示，探讨凸结构和序结构的联系，并将相关内容引入到模糊凸空间；最后，利用Q-范畴理论将集合范畴Set上的模糊拓扑结构和凸结构的松代数表示理论推广到基范畴Set↓L。本项研究属于拓扑、凸结构、范畴代数和序的交叉领域，融合模糊集、多值逻辑和enriched范畴等理论，为模糊拓扑和凸空间以及相关理论的研究提供新的视角、注入新的活力。

The theory of lax algebra is the synthetical development of algebra with respect to monad，relational algebra and enriched category. It plays important roles in the representations of special structures and it can reflect the algebraic properties of the structures well. The aim of this project is to study the monadic approaches to fuzzy topological structures and convex structures and its applications in them under the valued-lattice L is a commutative and unital quantale. Firstly, we will define the concept of fuzzy semiuniform convergence space and study its categorical properties as well as the regular completeness. We will give the representations of fuzzy topology and fuzzy semiuniform convergence space by lax algebras based on fuzzy filter monad and the identity monad, respectively. Secondly, we will define the concept of convex ideal and construct the convex ideal monad. We use the convex ideal monad to characterize convex spaces in two ways, one is by the Kleisli monoid, the other one is by the reflexive and transitive lax algebra. We also want to study the relationship between convex structures and preordered sets. Furthermore, we will generalize these results to fuzzy convex spaces. Finally, we use quantaloid-enriched category to extend the theory of representations of fuzzy topological structures and convex structures by lax algebras on base category Set to Set↓L.This project belongs to crossing field of topology,convex structure,lax algebra and order. It also compromises the theory of fuzzy set, many-valued logic and enriched-category. The results of this project will provide a new perspective of the research of fuzzy topology and fuzzy convex structure, and infuse new energies to them.