Perfect matroids over 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 $a⊞b$ 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.

19 pages

References in corpus (2)

• Field extensions, Derivations, and Matroids over Skew Hyperfields
• Classification of doubly distributive skew hyperfields and stringent hypergroups

Cited by in corpus (3)

• Classification of doubly distributive skew hyperfields and stringent hypergroups
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