Matroids over tropical extensions of tracts

TL;DR

This paper develops a unified framework for studying matroids over tropical extensions of algebraic structures called tracts, generalizing existing concepts like valuated and oriented matroids, and introduces new results on their properties and applications.

Contribution

It generalizes the correspondence between valuated and initial matroids to tropical extensions, and extends the theory to flag matroids, positroids, and enriched tropical linear spaces.

Findings

Characterization of $F[b3]$-matroids via initial matroids

Extension of circuit and covector descriptions to tropical extensions

Structure theorem for enriched tropical linear spaces

Abstract

A tract $F$ is an algebraic structure where multiplication is defined but addition is only partially defined. They were introduced by Baker and Bowler as a unified framework to study generalisations of matroids, including oriented and valuated matroids. A tropical extension $F[\Gamma]$ is a tract obtained by extending a tract $F$ by an ordered abelian group $\Gamma$. Key examples include the tropical hyperfield as a tropical extension of the Krasner hyperfield, and the signed tropical hyperfield as a tropical extension of the sign hyperfield. We study matroids over tropical extensions of tracts, including valuated matroids and oriented valuated matroids. We generalise the correspondence between valuated matroids and their initial matroids, showing that $M$ is an $F[\Gamma]$-matroid if and only if every initial matroid $M^u$ is an $F$-matroid. We also show analogous results for flag matroids and positroids over tropical extensions, utilising the circuit and covector descriptions we derive for $F[\Gamma]$-matroids. We conclude by studying images of linear spaces in enriched valuations, valuation maps enriched with additional data. These give rise to enriched tropical linear spaces, including signed tropical linear spaces as a key examples. As an application of our results, we prove a structure theorem for enriched tropical linear spaces, generalising the characterisation of projective tropical linear spaces of Brandt-Eur-Zhang.