2021/07/24 by Matthew Baker, Tianyi Zhang, Baker, Matthew +1
Computer Science · Mathematics · #Polynomial and algebraic computation #Commutative Algebra and Its Applications #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2107.11700
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 σ(F) with the same 3-term additive relations as F such that weak F-matroids coincide with strong σ(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.