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
In this post and its sequel, I’d like to explain a new perspective on matroid theory which was introduced in this recent preprint which I wrote with Nathan Bowler. The main observation is that by working with algebraic structures called hyperfields, in which addition is allowed to be multi-valued, linear subspaces, matroids, valuated matroids, and oriented matroids become special cases of a single general concept. In the process of explaining this observation, I also hope to advertise how natural hyperfields are, for example in the context of tropical geometry.
The notion of an algebraic structure in which addition is allowed to be multi-valued goes back to Frédéric Marty, who introduced hypergroups in 1934. Later on, in the mid-1950’s, Marc Krasner developed the theory of hyperrings and hyperfields in the context of approximating non-Archimedean fields, and in the 1990’s Murray Marshall explored connections to the theory of real spectra and spaces of orderings. For the most part, however, the theory of hyperstructures was largely ignored by the mathematical community at large until Connes and Consani started advocating its potential utility in connection with -geometry and the Riemann hypothesis. There now seems to be a reappraisal of sorts going on within the math community of the “bias” against multi-valued operations. Continue reading