Fusion rules for pastures and tracts

TL;DR

This paper introduces strongly fused tracts, a new class of algebraic structures that generalize existing concepts like partial fields and hyperfields, and proves their perfection property related to matroid theory.

Contribution

It defines strongly fused tracts, proves they are perfect, and establishes a construction linking weak and strong matroids over related tracts, generalizing previous results.

Findings

Strongly fused tracts are perfect.

Partial fields and stringent hyperfields are strongly fused.

A construction relates weak and strong matroids over different but related tracts.

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.