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arXiv · 1908.03420

Perfect matroids over hyperfields

Nathan Bowler·Rudi Pendavingh
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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.

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Nathan Bowler, Rudi Pendavingh. 2019-08-09. Perfect matroids over hyperfields. https://arxiv.org/abs/1908.03420

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