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arXiv · 2107.11700

Fusion rules for pastures and tracts

Matthew Baker·Tianyi Zhang
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Abstract

Baker and Bowler defined a category of algebraic objects called tracts which generalize both partial fields and hyperfields. They also defined a notion of weak and strong matroids over a tract $F$, and proved that if $F$ is perfect, meaning that $F$-vectors and $F$-covectors are orthogonal for every matroid over $F$, then the notions of weak and strong $F$-matroids coincide. We define the class of strongly fused tracts and prove that such tracts are perfect. We in fact prove a more general result which implies that given a tract $F$, there is a tract $\sigma (F)$ with the same 3-term additive relations as $F$ such that weak $F$-matroids coincide with strong $\sigma (F)$-matroids. We also show that both partial fields and stringent hyperfields are strongly fused; in this way, our criterion for perfection generalizes results of Baker-Bowler and Bowler-Pendavingh.

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Matthew Baker, Tianyi Zhang. 2021-07-24. Fusion rules for pastures and tracts. https://arxiv.org/abs/2107.11700

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