Matroids over hyperfields

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Plucker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.
Comment: 31 pages. v2: Fixed a few errors, added some new examples and remarks, added Theorem 4.17, removed the "Brief chronology" from v1. v3: Fixed some minor errors, streamlined the exposition. v4: Fixed a major error and added a co-author; see Section 1.7 for further details. v5: Added section 5 on doubly distributive hyperfields, Example 3.31 (due to Daniel Weissauer), and Theorem 3.8