Extensions of Matroids over Tracts and Doubly Distributive Partial Hyperfields

Matroids over tracts (Baker and Bowler, 2017) provide an algebraic framework simultaneously generalizing the notion of linear subspaces, matroids, oriented matroids, phased matroids, and some other ``matroids with extra structure." A single element extension of a matroid M over a tract is a matroid M˜ over a tract obtained from M by adding one more element. Crapo characterized single element extensions of ordinary matroids (Crapo, 1965), and Las Vergnas characterized single element extensions of oriented matroids in terms of single element extensions of their rank 2 contractions (Vergnas, 1978). We generalize their work for weak matroids over tracts when the tracts satisfy a necessary and sufficient algebraic property called Pathetic Cancellation Property. Doubly distributive partial hyperfields are special cases of tracts, which behave in many ways like fields. We find a similar characterization of single element extensions of strong matroids over doubly distributive partial hyperfields. We also provide a partial classification of doubly distributive partial hyperfields