Majorization precursors to supermodularity and subadditivity on the majorization lattice

2026-05-28

AI Summary

This work uncovers the structural origins of supermodularity and subadditivity on majorization lattices. By introducing, for the first time, two classes of majorization predecessor relations as sufficient conditions, it rigorously establishes that the sum of any concave functions is simultaneously strictly supermodular and strictly subadditive on such lattices. This framework unifies and strengthens the theory of information inequalities, and is successfully applied to Shannon entropy, Rényi entropy, and Tsallis entropy, thereby proving their strict subadditivity and supermodularity on majorization lattices. The results provide a novel theoretical foundation bridging information theory and lattice-ordered analysis.

Abstract

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.

Problem

• supermodularity
• subadditivity
• majorization lattice
• entropy
• concave functions

Innovation

• majorization lattice
• supermodularity
• subadditivity
• Tsallis entropy
• sum-concave functions