代数 domain 的两个笛卡尔闭满子范畴的概念表示

本项目拟利用范畴论方法为关系概念分析建立基本的数学模型，通过研究特殊形式背景的概念层次结构的序性质和拓扑性质，在范畴层面实现代数FS-domain 和代数 L-domain 的概念表示。具体地，研究内容包括：(1) 基于形式概念分析（FCA）在数据挖掘、知识发现中的应用，研究特殊形式背景范畴的极限、余极限、指数对象以及态射空间等基本结构的性质，进而揭示这些范畴的笛卡尔闭性。(2) 研究形式背景蕴含的新的概念模式，并讨论相应的概念层次结构的序性质，进而找到与代数 FS-domain 范畴和代数 L-domain范畴等价的特殊形式背景范畴。(3) 通过研究这些特殊形式背景的拓扑性质，在范畴层面研究它们与特殊 Sober 空间的相互联系。本项目不仅可以为 FCA 在对象-关系型数据分析中的应用提供数学基础，也可以为 Domain 结构的表示提供新途径，因此具有重要的理论意义和应用价值。

The aim of this project is to establish a fundamental mathematical model for relational concept analysis by means of category theory and provide the conceptual representations of algebraic FS-domains and algebraic L-domains by investigating the order-theoretical properties of the concept hierarchies and topological properties of special formal contexts. Particularly, the content includes: (1) based on the applications of formal concept analysis in data mining and knowledge discovery, studying the Cartesian closedess of the categories of special formal contexts by investigating the properties of their limits, colimits, exponential objects and morphism spaces; (2) studying new patterns of concepts and the order-theoretical properties of the associated concept hierarchies, and then find the categories of special formal contexts which are equivalent to those of algebraic FS-domains and algebraic L-domains; (3) studying the topological properties of special formal contexts and the interrelation between those special formal contexts and special Sober spaces. This project will provide not only a mathematical foundation for FCA-based applications in object-relational data analysis but also a new approach to representing Domain structures. Therefore, it has important theoretical significance and principle value.