Entropy and co-entropy of a covering approximation space

The notions of entropy and co-entropy associated to partitions have been generalized to coverings when Pawlak's rough set theory based on partitions has been extended to covering rough sets. Unfortunately, the monotonicities of entropy and co-entropy with respect to the standard partial order on partitions do not behave well in this generalization. Taking the coverings and the covering lower and upper approximation operations into account, we introduce a novel entropy and the corresponding co-entropy in this paper. The new entropy and co-entropy exhibit the expected monotonicity, provide a measure for the fineness of the pairs of the covering lower and upper approximation operations, and induce a quasi-order relation on coverings. We illustrate the theoretical development by the first, second, and third types of covering lower and upper approximation operations. (C) 2011 Elsevier Inc. All rights reserved.