Entropy of capacities on lattices and set systems

TL;DR

This paper introduces a new entropy measure for capacities on lattices, generalizing classical Shannon entropy and capacity entropy, with properties and examples illustrating its application.

Contribution

It proposes a novel entropy definition for capacities on lattices, extending classical entropy concepts to more general set systems.

Findings

The new entropy encompasses Shannon's entropy for probability measures.

It generalizes the entropy of classical capacities.

Properties and examples demonstrate its applicability.

Abstract

We propose a definition for the entropy of capacities defined on lattices. Classical capacities are monotone set functions and can be seen as a generalization of probability measures. Capacities on lattices address the general case where the family of subsets is not necessarily the Boolean lattice of all subsets. Our definition encompasses the classical definition of Shannon for probability measures, as well as the entropy of Marichal defined for classical capacities. Some properties and examples are given.