Title: Matroids over partial hyperstructures
Speaker: Matt Baker (Georgia Institute of Technology)
Abstract: Hyperfields are similar to their non-hyper counterparts, but addition is allowed to be multi-valued. We introduce a generalization of hyperfields called tracts, which also include partial fields in the sense of Semple and Whittle. After giving some examples, we present a simultaneous generalization of the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. We call such objects matroids over tracts. In fact, there are (at least) two natural notions of matroids over a tract F, which we call weak and strong F-matroids. We give different ``cryptomorphic'' axiom systems for such matroids, discuss duality theory, and present sufficient conditions which guarantee that the notions of weak and strong F-matroids coincide (for a given tract F).
Seminar URL: https://research.math.osu.edu/agseminar/