模糊超代数的若干问题研究

Fuzzy hyperalgebra is one of the hot spots in the research community, and the definition of which is the basis of it being accepted and widely applied. However, the concept of fuzzy hyperalgebraic structure in general is just a generalization of hyperalgebraic structure, which is still lack of theoretical basis, and the limitation of this kind of definition also makes it difficult to introduce some definitions of algebra (hyperalgebra) into fuzzy hyperalgebra. To solve the above problems, this research attempts to apply the theory of falling shadow into the study of fuzzy hyperalgebraic structure to develop the theoretical basis of the definition of fuzzy hyperalgebraic structure in general, presents a new definition for fuzzy hyperalgebraic structure from another viewpoint, and investigate the structure characteristics and properties of this kind of fuzzy hyperalgebraic structure and establish its category theory. The aim of this research is to further enrich the theory of fuzzy hyperalgebra and try to provide theoretical foundation for further application of fuzzy hyperalgebra.

模糊超代数是当今国际研究的热点问题之一，其定义的合理性是其被接受并得到应用的基础。而目前一般意义上的模糊超代数结构的定义是经典情形的形式推广，缺乏理论依据。此外，一般意义上的模糊超代数结构的定义本身也存在一定的局限性，导致很难将经典（超）代数学中的一些概念引入其中。鉴于此，本项目拟用落影表现理论来研究模糊超代数，探讨一般意义上的模糊超代数结构定义的理论依据；拟从另一角度重新考虑模糊超代数结构的定义，定义一种新型的模糊超代数结构，研究其与一般意义上的模糊超代数结构及经典超代数结构间的相互关系，探讨其结构特征和性质，尝试建立该类模糊超代数的范畴体系。目的是进一步完善模糊超代数理论，为模糊超代数的可能应用奠定理论基础。

整体来说，本项目较好的完成了预期目标。基于目前代数超结构学与模糊数学相结合的方法，主要开展了以下两个方面的研究：（一）深入研究了利用模糊超运算定义的L-模糊Krasner-超环的纵深问题，探讨了其上的各类代数结构及其格性质。（二）推广了目前各类模糊代数超结构的定义，定义并研究了一类更具一般性的模糊（软）n-序超群（超环），并初步探讨了落影表现理论在该类模糊超代数的应用。