Foundations of Matroids
Matroid theorists are interested in questions concerning representability of matroids over fields. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the universal partial field of a matroid, which governs the representations of over all partial fields. Unfortunately, most matroids are not representable over any partial field, and in this case, the universal partial field is not defined. Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the foundation of a matroid; it is always well-defined. The foundation is a type of algebraic object which we call a pasture; pastures include both hyperfields and partial fields. As a particular application of this point of view, I will explain the classification of all possible foundations for matroids having no minor isomorphic to U(2,5) or U(3,5). Among other things, this provides a short and conceptual proof of the 1997 theorem of Lee and Scobee which says that a matroid is both ternary and orientable if and only if it is dyadic.
