Amalgams of matroids, fibre products and tropical graph correspondences

TL;DR

This paper establishes a connection between matroid amalgams and tropical geometry by showing the existence of proper amalgams corresponds to positivity in tropical fibre products of Bergman fans, introducing tropical correspondences and functorial graph constructions.

Contribution

It introduces tropical correspondences between Bergman fans, generalizes Bergman fans to Flag fans, and links matroid amalgams with tropical fibre product positivity.

Findings

Proper amalgam of matroids exists iff tropical fibre product of Bergman fans is positive.

Introduces tropical correspondences as tropical subcycles in product fans.

Proves graph construction as a functor for lattice covering maps.

Abstract

We prove that the proper amalgam of matroids $M_1$ and $M_2$ along their common restriction $N$ exists if and only if the tropical fibre product of Bergman fans ${B(M_1) \times_{B(N)} B(M_2)}$ is positive. We introduce tropical correspondences between Bergman fans as tropical subcycles in their product, similar to correspondences in algebraic geometry, and define a "graph correspondence" of the map of lattices. We prove that graph construction is a functor for the "covering" maps of lattices, exploiting a generalization of Bergman fan which we call a "Flag fan".