偏概率度量空间的模糊集方法

偏度量和概率度量是基于各自应用背景对经典度量的两种不同推广，在理论和实际中都有着重要的应用。 本项目的目的是将把概率度量、偏度量和序理论相结合，引入更宽泛的偏概率度量，借助模糊集和多值逻辑的思想建立多值的偏概率度量空间理论。 首先合理地给出偏概率度量空间的定义，讨论概率度量，偏度量和偏概率度量的关系。我们知道多值拓扑才是研究概率度量空间拓扑化的工具，然后，在偏概率度量空间中给出导出多值拓扑结构的方式，研究偏概率度量的多值拓扑化问题，并建立偏概率度量空间和模糊化一致空间中用程度定义的多值收敛度理论、完备度理论及其应用。最后，在带有序结构的论域上，将序和偏概率度量结合从enriched范畴的角度来研究偏概率度量空间。

Partial metric and probabilistic metric are two kinds of generalization of classical metric based on different application background, and play important role in mathematical theory and application. The aim of this project is to give the definition of partially probabilistic metric based on probabilistic metric, partial metric and order, and use the idea of fuzzy sets and many-valued logic to study many-valued theory of partially probabilistic metric spaces. Firstly, we will give the concept of partially probabilistic metric, and study the relationship between partially probabilistic metric and probabilistic metric as well as the relationship between partially probabilistic metric and partial metric. As we know many-valued topology is the key tool to study topologization of probabilistic metric spaces. Secondly, partially probabilistic metric will be endowed with many-valued topological stuctures and the topologization of partially probabilistic metric spaces. In this part, we will also establish the convergence degree theory, completeness degree theory and their applications in partially probabilistic metric spaces and fuzzifying uniform spaces. Finally, we will study partially probabilistic metric spaces in the framework of enriched categoies when there is order-stucture on the universal set.