In Part I of this post, we defined hyperrings and hyperfields, gave some key examples, and introduced matroids over (doubly distributive) hyperfields in terms of Grassman-Plücker functions. If is a finite set and is a field, we saw that a -matroid on is the same thing as a linear subspace of , and if is the Krasner hyperfield then a -matroid on is the same thing as a matroid in the usual sense. Matroids over the hyperfield of signs are the same thing as oriented matroids, and matroids over the tropical hyperfield are the same thing as valuated matroids. In this post we will give some “cryptomorphic” axiomatizations of matroids over hyperfields, discuss duality theory, and then discuss the category of hyperrings in a bit more detail. Continue reading