Submodular functions, generalized permutahedra, conforming preorders, and cointeracting bialgebras

To a submodular function we define a class of preorders, conforming preorders. A submodular function $z$ corresponds to a generalized permutahedron $\Pi(z)$. We show the faces of $\Pi(z)$ are in bijection with the conforming preorders. The face poset structure of $\Pi(z)$ induces two order relations $\lhd$ and $\blacktriangleleft$ on conforming preorder, and we investigate their properties. Ardila and Aguiar introduced a Hopf monoid of submodular functions/generalized permutahedra. We show there is a cointeracting bimonoid of modular functions. By recent theory of L.Foissy this associates a canonical polynomial to any submodular function.