Weakly submodular rank functions, supermatroids, and the flat lattice of a distributive supermatroid

Distributive supermatroids generalize matroids to partially ordered sets. Completing earlier work of Barnabei, Nicoletti and Pezzoli we characterize the lattice of flats of a distributive supermatroid. For the prominent special case of a polymatroid the description of the flat lattice is particularly simple. Large portions of the proofs reduce to properties of weakly submodular rank functions. The latter are also investigated for their own sake, and some new results on general supermatroids are derived.