We establish two structural majorization relations, which we call precursors, underlying the properties of supermodularity and subadditivity on the lattice induced by majorization. These are precursors in that they immediately imply that all sums of concave functions, which we dub sum-concave functions, are supermodular and subadditive on the majorization lattice. Using these majorization relations, we then show the supermodularity and subadditivity (in the lattice-theoretic sense) of Tsallis entropies (for all $α$) and Rényi entropies (for all $α> 1$), also recovering these properties for the Shannon entropy in the process. We further strengthen these inequalities, showing that: (i) all these entropic functionals are strictly subadditive on the majorization lattice; (ii) Tsallis entropies (and therefore the Shannon entropy as well) are strictly supermodular on the majorization lattice.

翻译：我们建立了两种结构性的majorization关系（称为前驱），它们构成了由majorization诱导的格上超模性与次可加性的基础。这些关系之所以被称为前驱，是因为它们直接蕴含：所有凹函数的和（我们称之为凹和函数）在majorization格上既是超模的又是次可加的。利用这些majorization关系，我们进一步证明了Tsallis熵（对所有$\alpha$）与Rényi熵（对所有$\alpha>1$）在格论意义上的超模性与次可加性，并在过程中重新推导了Shannon熵的这些性质。我们进一步强化了这些不等式，证明：(i) 所有这些熵泛函在majorization格上都是严格次可加的；(ii) Tsallis熵（因此也包括Shannon熵）在majorization格上具有严格超模性。