Single-element extensions of matroids over skew tracts

TL;DR

This paper investigates the conditions under which single-element extensions of matroids over skew tracts can be characterized, generalizing classical results to a broader algebraic framework involving skew hyperfields.

Contribution

It introduces the Pathetic Cancellation condition on skew tracts and characterizes single-element extensions of strong matroids over skew hyperfields.

Findings

Pathetic Cancellation condition is necessary and sufficient for generalizing single-element extension results.

Characterization of extensions over stringent skew hyperfields.

Extension results unify various matroid theories under skew tract framework.

Abstract

Matroids over skew tracts provide an algebraic framework simultaneously generalizing the notions of linear subspaces, matroids, oriented matroids, phased matroids, and some other ``matroids with extra structure". A single-element extension of a matroid $\mathcal{M}$ over a skew tract $T$ is a matroid $\widetilde{\mathcal{M}}$ over $T$ obtained from $\mathcal{M}$ by adding one more element. Crapo characterized single-element extensions of ordinary matroids, and Las Vergnas characterized single-element extensions of oriented matroids, in terms of single-element extensions of their rank 2 contractions. The results of Crapo and Las Vergnas do not generalize to matroids over skew tracts, but we will show a necessary and sufficient condition on skew tracts, called Pathetic Cancellation, such that the result can generalize to weak matroids over skew tracts. Stringent skew hyperfields are a special case of skew tracts which behave in many ways like skew fields. We find a characterization of single-element extensions of strong matroids over stringent skew hyperfields.