Characterization for entropy and solution of set function on lattice

We propose a definition for the entropy of a monotone set function defined on a lattice which are not necessarily the whole power set, but satisfy the condition of regularity. Our definition encompasses the classical definition of Shannon for probability measures, as well as the definition of Marichal for classical fuzzy measures and may have applicability to most fuzzy measures which appear in applications. We give also an axiomatization of this entropy. This axiomatization is in the spirit of Faddeev's axiomatization for the classical Shannon entropy. After that, using same idea we introduce a generalization of the Shapley value for a set function defined on a lattice and give two types of necessary and sufficient conditions.