Abstract: In Pawlak rough set model, the lower and upper approximations possess many mathematical properties, including the dual property. However, some of them may not hold in generalized rough set models. So, Yao defined new lower and upper approximations with respect to general binary relations based on right neighborhoods. In Yao’s definitions, the lower (upper) approximation was defined by the similar way with classical rough set, whereas the upper (lower) approximation was defined as the dual of the lower (upper) approximation. To preserve some properties in Pawlak’s model, this paper dually defines two types of lower and upper approximations based on the maximal tolerance classes in tolerance rough set model, and discusses their properties. Compared with the properties of the approximations defined in original tolerance rough set model, the new approximations proposed in this paper retain most of them, and satisfy some new properties.

Key words: rough set, tolerance relation, maximal tolerance class, approximation

摘要： 在经典粗糙集模型中，上、下近似满足很多好的数学性质，其中包括对偶性。在一般粗糙集模型中，这些性质中的某一些不再被满足。Yao给出了在一般粗糙集模型中定义上、下近似的新思路，即首先利用对象的邻域定义下近似（上近似），然后再将上近似（下近似）定义为该下近似（上近似）的对偶。为保留经典粗糙集模型中一些好的数学性质，根据Yao的方法，在相容粗糙集模型中利用最大相容类对偶地定义了两类新的上、下近似概念，并讨论了它们的性质。对比发现，这两类新的上、下近似保留了大部分初始相容粗糙集模型中上、下近似所满足的性质，且有新的性质成立。

关键词: 粗糙集, 相容关系, 最大相容类, 近似