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

TL;DR

This paper explores the deep connections between submodular functions, generalized permutahedra, and conforming preorders, revealing new bijections, order relations, and algebraic structures like bimonoids and double bialgebras.

Contribution

It introduces conforming preorders linked to submodular functions and generalized permutahedra, and uncovers a novel bimonoid structure of modular functions cointeracting in a unique way.

Findings

Faces of generalized permutahedra correspond to conforming preorders.

Two order relations on preorders relate to face poset structures.

Existence of a canonical polynomial associated with submodular functions.

Abstract

Submodular functions $z$ defined on the power set of a finite set are in bijection with generalized permutahedra $\egp(z)$. To any such $z$ we define a class of preorders, {\it conforming} preorders. We show the faces of $\egp(z)$ and the conforming preorders are in bijection. We investigate in detail this interplay between submodular functions and generalized permutahedra on one side, and conforming preorders on the other side, with many examples. In particular, the face poset structure of $\egp(z)$ correspond to two order relations $\lhd$ and $\btl$ on preorders, and we investigate their properties. Ardila and Aguiar \cite{AA2017} introduced a Hopf monoid of submodular functions/generalized permutahedra. We show there is a bimonoid of modular functions cointeracting in a non-standard way. By recent theory of L.Foissy \cite{Fo2022}, on double bialgebras we get a canonical polynomial associated to any submodular function.