Matroids over tropical extensions of tracts
Ben Smith

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 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 is a tract obtained by extending a tract by an ordered abelian group . 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 is an -matroid if and only if every initial matroid is an -matroid. We also show analogous results for flag…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics
