A fuzzy relation information system may be viewed as an information system with fuzzy relations. Uncertainty measurement is a critical evaluating tool. This paper investigates uncertainty measurement for a fuzzy relation information system. The concept of information structures in a fuzzy relation information system is first described by using set vectors. Then, dependence between information structures in a fuzzy relation information system is given. Next, the axiom definition of the granularity measurement of the uncertainty for fuzzy relation information systems is proposed by means of its information structures. Based upon this axiom definition, information granulation and rough entropy in a fuzzy relation information system are proposed. Moreover, information entropy, information amount, joint entropy, and condition entropy in a fuzzy relation information system are also considered. To show the feasibility of the proposed measures for uncertainty of a fuzzy relation information system, effectiveness analysis is conducted from the angle of statistics. Finally, characterizations of fuzzy relation information systems under a compatible homomorphism are obtained. These results will be helpful for understanding the essence of uncertainty in a fuzzy relation information system.

中文翻译：

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模糊关系信息系统的不确定性测量


模糊关系信息系统可以看作是具有模糊关系的信息系统。不确定性测量是一个重要的评估工具。本文研究了模糊关系信息系统的不确定性测量。首先使用集合向量来描述模糊关系信息系统中信息结构的概念。然后，给出了模糊关系信息系统中信息结构之间的依赖关系。其次，利用模糊关系信息系统的信息结构，给出了模糊关系信息系统不确定性粒度度量的公理定义。基于该公理定义，提出了模糊关系信息系统中的信息粒度和粗糙熵。此外，还考虑了模糊关系信息系统中的信息熵、信息量、联合熵和条件熵。为了证明所提出的模糊关系信息系统不确定性测量方法的可行性，从统计学的角度进行了有效性分析。最后，得到了兼容同态下模糊关系信息系统的表征。这些结果将有助于理解模糊关系信息系统中不确定性的本质。