Georgia Institute of Technology College of Sciences

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects which we call pastures; they generalize both hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. We then define matroids over pastures; in fact, there are at least two natural notions of matroid in this general context, which we call weak and strong matroids. We present ``cryptomorphic'’ descriptions of each kind of matroid. To a (classical) rank-$r$ matroid $M$ on $E$, we can associate a universal pasture (resp. weak universal pasture) $k_M$ (resp. $k_M^w$). We show that morphisms from the universal pasture (resp. weak universal pasture) of $M$ to a pasture $F$ are canonically in bijection with strong (resp. weak) representations of $M$ over $F$. Similarly, the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios'', which we call the foundation of $M$, parametrizes rescaling classes of weak $F$-matroid structures on $M$. As a sample application of these considerations, we give a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.