Fusion rules for pastures and 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 $\ddot{I}ƒ(F)$ with the same 3-term additive relations as $F$ such that weak $F$-matroids coincide with strong $\ddot{I}ƒ(F)$-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.