Perfect matroids over hyperfields

TL;DR

This paper explores the structure of valuated matroids over hyperfields, establishing conditions under which weak matroids are strong and generalizing vector axioms to a broader class of hyperfields.

Contribution

It introduces the concept of stringent hyperfields and demonstrates their role in characterizing weak and strong matroids, extending axioms beyond traditional frameworks.

Findings

Weak matroids over stringent hyperfields are orthogonal to their covectors.

Weak matroids over such hyperfields are equivalent to strong matroids.

Vector axioms are generalized for matroids over stringent skew hyperfields.

Abstract

We investigate valuated matroids with an additional algebraic structure on their residue matroids. We encode the structure in terms of representability over stringent hyperfields. A hyperfield $H$ is {\em stringent} if $a\boxplus b$ is a singleton unless $a=-b$, for all $a,b\in H$. By a construction of Marc Krasner, each valued field gives rise to a stringent hyperfield. We show that if $H$ is a stringent skew hyperfield, then the vectors of any weak matroid over $H$ are orthogonal to its covectors, and we deduce that weak matroids over $H$ are strong matroids over $H$. Also, we present vector axioms for matroids over stringent skew hyperfields which generalize the vector axioms for oriented matroids and valuated matroids.