中文摘要：

在文献[16]基础上,进一步将模糊粒度空间推广到更一般地模糊等价关系上,研究了模糊粒度空间的性质,主要获得了3个结论。首先,引入了有序的等价关系集的概念,给出了下列的四个命题是等价的：（1）给定一个模糊等价关系;（2）给定一个等腰归一化伪距离;（3）给定一个有序的粒度空间;（4）给定一个有序的等价关系集。第二,通过模糊等价关系诱导的等腰归一化伪距离的投影距离和扩展距离,建立了模糊粒度空间上的距离,即是等腰归一化距离,并且给出了模糊粒度空间上距离度量的动态性质研究。最后,给出了模糊粒度空间与模糊等价关系之间的序关系,即它们的序是一致的。这些研究工作进一步完善了模糊粒度空间的理论,为模糊粒度计算提供了更为直观的数学理论和工具。

英文摘要：

On the basis of Reference [16], we introduced fuzzy granular space into general fuzzy equivalence relations, studied its properties, and get three main conclusions. Firstly, the ordered equivalence relations set is introduced, and four propositions are equivalent as the follows： （1） A fuzzy equivalence relation is given; （2） An equicrural normalized pseudo-metric is given; （3） An ordered granular space is given; （4） An ordered equivalence relations set is given. Secondly, by introducing the projective metric and extending metric of an equicrural normalized pseudo-metric deriving from a fuzzy equivalence relation, the metric on fuzzy granular space is determined, which it is an equicrural normalized metric, and its dynamic properties are discussed. Finally, the relationship between order of fuzzy granular space and one of fuzzy equivalence relation is discussed, which their order are same. There conclusions prefect the fuzzy granular space theory, and further provide a direct and geometric interpretation for granular computing, and will help us for deeper understanding the essence of granular procedure.