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

Gunnar Fløystad, Dominique Manchon

TL;DR

The paper builds a comprehensive framework tying together finite preorders/topologies, submodular functions, and (extended) generalized permutahedra through a Galois-theoretic lens. It introduces conforming preorders as the natural combinatorial avatars of faces of $\Pi(z)$ and proves a bijection between these preorders and faces of $\Pi(z)$, with two central relations $\lhd$ and $\blacktriangleleft$ governing subdivisions and contractions. It then develops a bimonoid/cointeraction theory for submodular and modular functions, including a morphism to modular data and a translation to canonical polynomials via Foissy’s double bialgebra, culminating in Ehrhart-polynomial-type invariants. The work unifies braid-fan geometry, nested-set/nestohedra combinatorics, and matroid theory within a single algebraic-combinatorial apparatus, yielding new perspectives on face lattices, normal fans, and associated polynomials with potential for broad applications in optimization and geometry.

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.

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

TL;DR

The paper builds a comprehensive framework tying together finite preorders/topologies, submodular functions, and (extended) generalized permutahedra through a Galois-theoretic lens. It introduces conforming preorders as the natural combinatorial avatars of faces of and proves a bijection between these preorders and faces of , with two central relations and governing subdivisions and contractions. It then develops a bimonoid/cointeraction theory for submodular and modular functions, including a morphism to modular data and a translation to canonical polynomials via Foissy’s double bialgebra, culminating in Ehrhart-polynomial-type invariants. The work unifies braid-fan geometry, nested-set/nestohedra combinatorics, and matroid theory within a single algebraic-combinatorial apparatus, yielding new perspectives on face lattices, normal fans, and associated polynomials with potential for broad applications in optimization and geometry.

Abstract

Submodular functions defined on the power set of a finite set are in bijection with generalized permutahedra . To any such we define a class of preorders, {\it conforming} preorders. We show the faces of 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 correspond to two order relations and 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.
Paper Structure (44 sections, 49 theorems, 214 equations, 16 figures)

This paper contains 44 sections, 49 theorems, 214 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Preorders, topologies and submodular functions
  3. Preorders and topologies
  4. Submodular functions
  5. From submodular functions to topologies and preorders
  6. Matroids and polymatroids
  7. Extended generalized permutahedra
  8. Definition and examples
  9. Classes of extended generalized permutahedra
  10. Construction by Minkowski sums
  11. Building sets and nestohedra
  12. The braid fan
  13. Subdivisions and contractions of preorders
  14. A Galois correspondence on ${\mathrm {Pre}}(X)$
  15. Properties of the relations $\lhd$ and $\blacktriangleleft$
  16. ...and 29 more sections

Key Result

Lemma 2.3

Any extended Boolean function $z \in {\rm BF}\,[I]$ is uniquely a product of indecomposable extended Boolean functions.

Figures (16)

  • Figure 1: Two-dimensional permutahedron
  • Figure 2: One-dimensional permutahedron
  • Figure 3: Fan of one-dimensional permutahedron $\rightsquigarrow$ reduced fan
  • Figure 4: The extended generalized permutahedron ${\Pi}({\mathrm {low}}_P)$
  • Figure 5: Fan and reduced fan of ${\Pi}({\mathrm {low}}_P)$
  • ...and 11 more figures

Theorems & Definitions (135)

  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 3.1
  • Lemma 3.2
  • proof
  • ...and 125 more