拓扑EQ-代数的结构研究

EQ-logic is a kind of higher-order fuzzy logics, EQ-algebras are its corresponding algebras of truth values, and topological EQ-algebras are topological logic algebras in which algebraic structures and topological structures come in contact most naturally. The main aim of this project is to reveal the essence of topological EQ-algebras based on the filter structures, para-topological structures and completion structures of topological EQ-algebras. The main research contents are as follows. (1) We develop the filter theory of EQ-algebras to solve the proplems about filters. (2) We analyse the structures of topological EQ-algebras by using para- meet - topological EQ-algebras, para- multiplication - topological EQ-algebras and para - fuzzy equality - topological EQ-algebras. (3) We construct the completion of topological EQ-algebras by using the convergence theory of topological EQ-algebras. Based on this project, the deep fusion of the order structures, algebraic structures and topological structures on EQ-algebras will be obtained. This project will make a beneficial exploration for the research of other logic algebras and lay some algebraic foundations for higher-order fuzzy logics.

EQ-逻辑是一类高阶模糊逻辑，EQ-代数是其对应的真值代数，而拓扑EQ-代数是EQ-代数中代数结构与拓扑结构有机融合的拓扑逻辑代数。本项目基于拓扑EQ-代数的滤子结构、仿拓扑结构以及完备化结构的研究揭示拓扑EQ-代数的本质。主要研究内容包括：(1)发展EQ-代数的滤子理论，解决目前滤子方面存在的问题；(2)利用仿-交-拓扑EQ-代数、仿-乘-拓扑EQ-代数以及仿-模糊相等-拓扑EQ-代数剖析拓扑EQ-代数的结构；(3)通过对拓扑EQ-代数收敛理论的研究，构造拓扑EQ-代数的完备化。本项目拟实现EQ-代数上序结构、代数结构和拓扑结构的深度融合，同时为其他逻辑代数的研究做有益探索，也为高阶模糊逻辑奠定一定代数基础。

EQ-代数是高阶模糊型逻辑的代数语义，拓扑EQ-代数是EQ-代数的代数结构和拓扑结构的有机组合。本项目通过深入分析EQ-代数的结构得到的主要研究进展、重要结果概括如下：1.完善了EQ-代数的滤子理论，建立了EQ-代数的谱理论；2.在EQ-代数的真子类剩余格中深入分析了代数结构和拓扑结构之间的相容性，解决了拓扑逻辑代数的两个公开问题；3.在EQ-代数的相关逻辑代数上研究了态的存在性和逻辑结构。通过本项目的研究得到了处理一般代数的方法论，即零维线性化方法，此方法在理论上具有广泛的应用性，特别在研究拓扑逻辑代数时它和Stone对偶理论具有异曲同工之妙。截止目前已经在《Fuzzy Sets and Systems》(SCI 一区Top期刊)、《Soft Computing》、《Archive for Mathematical Logic》、《Journal of Logic and Computation》、《Matematica Slovaca》SCI期刊发表学术论文5篇，目前正在培养硕士研究生3人。