Entropies and Co-Entropies of Coverings with Application to Incomplete Information Systems

Different generalizations to the case of coverings of the standard approach to entropy applied to partitions of a finite universe X are explored. In the first approach any covering is represented by an identity resolution of fuzzy sets on X and a corresponding probability distribution with associated entropy is defined. A second approach is based on a probability distribution generated by the covering normalizing the standard counting measure. Finally, the extension to a generic covering of the Liang–Xu approach to entropy is investigated, both from the “global” and the “local” point of view. For each of these three possible entropies the complementary entropy (or co–entropy) is defined showing in particular that the Liang–Xu entropy is a co–entropy.