模糊数度量的单值函数表示

Monotropic Function Representation of Metrics on Fuzzy-number Space

基于一个具体的三角模糊数作为结构元,给出了模糊数与定义于[-1,1]上的单调递增有界且上半连续的函数构成的集合之间的一一对应关系,从而得到了模糊数的单值函数表示。在上述表示基础上,给出了模糊数空间上的各种度量如上确界度量、Lp度量、sendograph度量的单值函数表现,从而将对模糊数的度量(拓扑)性质的研究完全转化为对普通单调递增有界且上半连续的函数空间中相应性质的研究。

By taking a specific triangular fuzzy number as structural element, this paper gives a bijection between the set of all fuzzy numbers and the set of all bounded, monotonic increasing and upper semi-continuous functions defined on [- 1, 1], thus get a monotropic function representation of fuzzy number. On the basis of above representation, this paper gives the monotropic function representation of various metrics on fuzzy number space, such as the supremum metric, the Lp metrics, and the sendograph metric, so as to fully translate the study on metric （topology） properties of fuzzy numbers into the study on the corresponding properties on the space of all bounded, monotonic increasing and upper semi-continuous functions.