On Non-pointwise Entropies of Coverings: Relationship with Anti-monotonicity

Some approaches to the entropy of coverings, generalizing the case of the partition entropy, and some definitions of orderings and quasi-orderings of coverings as extensions to the covering context of various formulations of the standard order on partitions. Unfortunately, we here show that the problem of anti-monotonicity of non-pointwise entropy of coverings is still open. On the other side, we will illustrate an approach to generate a partition from a covering; we will then show how we can indirectly compare the entropies of two coverings, that are in a certain order relation, by comparing the entropies of the induced partitions.

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