序结构赋值范畴与层结构提升范畴之间的同构关系  

Isomorphic Relation between Categories of Ordered Evaluations and Shifting Categories of Sheaf Structures

作　　者：哈书敏 汤建钢[1] 罗懋康[2] 

机构地区：[1]伊犁师范学院数学系 [2]四川大学数学学院

出　　处：《模糊系统与数学》2016年第3期9-18,共10页Fuzzy Systems and Mathematics

基　　金：伊犁师范学院校级研究生2015年度科研创新项目(2015YSY020)

摘　　要：序结构赋值,是将数学对象赋以序化层次结构(ordered stratifications)从而可在一个包含额外的序结构维度的空间(a space including an extra ordered dimension)中以更为丰富的工具和方法对其进行研究的有效方法,无论在纯数学还是在应用数学乃至各种具体应用中,均有广泛的运用。从基础数学中以各种不同赋值方式处理真假判断的逻辑,到应用数学中以不同赋值方式处理不确定性理论中基于由因果律破缺造成的随机性和由排中律破缺造成的模糊性的概率论和模糊理论,均为典型的序结构赋值。另一方面,局部性质与整体性质之间的关系永远是数学研究的中心内容之一,而层结构则是经典数学中从范畴层次处理这两种性质之间关系的经典理论和方法。对于这两种不同需求和不同处理方法,我们通过构造其间的同构关系而证明了它们之间可以相互等价转化,从而使得这两种结构和方法可以同时协调运用于同一数学对象的研究之中。Ordered evaluation is an effective methods to study mathematical objects by evaluating them with a proper structure of ordered stratifications, such that these objects can be analyzed and connected with more enriched tools and methods in a space including an extra ordered dimension. This method has very wide use in pure mathematics and applied mathematics even various concrete applications. From kinds of logic in pure mathematics for disposing true/false judgements with various modes of evaluation, to probability theory and fuzzy set theory in applied mathematics based on randomicity due to the breach of the Law of Causation, and fuzziness due to the breach of the Law of Excluded Middle, respectively, all are typical ordered evaluations. On the other hand, relations between local properties and global properties is always one of the central contents of mathematical research and in the meantime, sheaf structure is a classical theory and method in classical mathematics to dispose the relations between these two kinds of properties in the level of categories. To these two kinds of different requirements and disposing methods, we prove that they can be equivalently transformed to each other by constructing the isomorphic relations between them, and then make these two kinds of structures and methods can be harmoniously applied in the study of the same mathematical objects.

关 键 词：序结构 层结构 赋值范畴 提升范畴 同构关系 

分 类 号：O153[理学—数学]