Information Entropy and Granulation Co-Entropy of Partitions and Coverings: A Summary

Some approaches to the covering information entropy and some definitions of orderings and quasi–orderings of coverings will be described, generalizing the case of the partition entropy and ordering. The aim is to extend to covering the general result of anti–tonicity (strictly decreasing monotonicity) of partition entropy. In particular an entropy in the case of incomplete information systems is discussed, with the expected anti-tonicity result, making use of a partial partition strategy in which the missing information is treated as a peculiar value of the system. On the other side, an approach to generate a partition from a covering is illustrated. In particular, if we have a covering γ coarser than another covering δ with respect to a certain quasi order relation on coverings, the induced partition π(γ) results to be coarser than π(δ) with respect to the standard partial ordering on partitions. Thus, one can compare the two coverings through the entropies of the induced partitions.