Relationships between fuzzy approximation spaces and their uncertainty measures.

• Equality, dependence and independence between FA -spaces are proposed, and dependence and independence are characterized by the inclusion degree. • Distance between FA -spaces is considered from the angle of separation. • Algebraic, mapping and lattice characterizations of FA -spaces are obtained. • Measuring uncertainty of FA -spaces is investigated. • To evaluate the performance of the proposed measures for uncertainty of FA -spaces two examples are employed to do effectiveness analysis from the viewpoint of statistics. Fuzzy approximation spaces (FA -spaces) can be seen as approximation spaces (A -spaces) in a fuzzy environment. This article studies relationships between FA -spaces and their uncertainty measures. Dependence, partial dependence and independence between FA -spaces, as characterized by the inclusion degree, are first proposed. Then, the distance between FA -spaces is considered in terms of the angle of difference. Next, algebraic, map and lattice features of FA -spaces are obtained. Finally, the uncertainty of FA -spaces is measured by using the dependence between them, an application is given, and to assess the performance of the presented measures, two examples are used to perform effectiveness analysis by means of elementary statistical methods. [ABSTRACT FROM AUTHOR]