关于一类模糊代数及其模糊凸性的研究

凸空间理论是通过凸集的基本性质抽象而来的数学分支，许多经典代数结构中都具有凸性，将凸空间理论与模糊数学理论相结合，产生了模糊凸空间理论。模糊代数是模糊数学理论研究最为活跃的研究方向之一，但常见的模糊代数的研究受到经典代数的制约，且传统的模糊代数很难找到与模糊凸空间之间的联系，本项目的研究旨在解决上述两个问题，一方面我们通过模糊运算给出一类新的模糊代数结构的公理化定义，注重元素间程度上的对应关系，称之为M-hazy代数结构。刻画各种M-hazy代数的公理化定义，包括M-hazy群、M-hazy模格等。另一方面，我们通过此类新的模糊代数分别诱导出不同的M-模糊化凸空间，并研究每一种M-模糊化凸空间的性质，如arity性、分离性等。本项目的研究将为凸空间理论及其代数结构的研究提供新的思路和方法，将为不确定性理论与信息领域交叉学科提供理论基础。

The theory of convex spaces is a mathematical branch through abstracting the basic properties of convex sets. Convexity exists in many classical algebraic structures. The theory of convex spaces and fuzzy mathematics theory combine to produce the theory of fuzzy convex spaces. Fuzzy algebras are one of the hottest topics in the fuzzy mathematics theory. However, the study of general fuzzy algebras is restricted by classical algebras. And it is difficult to find the connection between traditional fuzzy algebras and fuzzy convex spaces. The research of this project is to solve the above two problems. On the one hand, we give a new axiomatic definition of fuzzy algebraic structures based on hazy binary operations, and pay attention to the degree relationship between elements, and name it M-hazy algebra. Give the axiomatic definition of different kinds of M-hazy algebras, including M-hazy groups, M-hazy modular lattice and others. On the other hand, different M-fuzzifying convex spaces are induced by these new fuzzy algebras. And investigates the properties of M-hazy algebras and related M-fuzzifying convex spaces, such as arity and separation properties. The research of this project will provide new thinking and method for convex space theories and algebras and to provide theoretical base for uncertainty theories as well as interdisciplinary subject related information field.