Universal partial hyperfields of matroids and their prespaces of orderings

Abstract

We associate a partial hyperfield 𝕌⁽⁰⁾(M) with every matroid M by defining an addition on the elements of its inner Tutte group with an additional zero element such that M is representable over 𝕌⁽⁰⁾(M), and every representation of M over a partial hyperfield F factors over the representation of M over 𝕌⁽⁰⁾(M). We investigate the relationship between 𝕌⁽⁰⁾(M) and 𝕌⁽⁰⁾(N) for minors N of M and prove that 𝕌⁽⁰⁾(M) is the coproduct of 𝕌⁽⁰⁾(Mᵢ), i=1,…,k where M₁,…,Mₖ are the connected components of M. Further, we examine the possible non-trivial decompositions of 𝕌⁽⁰⁾(M) as a coproduct and present sufficient geometrical conditions under which no such decomposition exists. We develop an Artin-Schreier-Theory for partial hyperfields and show that the orderings of a partial hyperfield form a prespace of orderings, which is in general not a space of orderings in the sense of Marshall, even for the partial hyperfield 𝕌⁽⁰⁾(M) of a matroid M.

Moreover, we provide examples of matroids M for which 𝕌⁽⁰⁾(M) is a hyperfield and its prespace of orderings is a space of orderings in the sense of Marshall, including affine space of dimension at least 3 and affine translation planes whose kernel contains at least four elements, for which the inner Tutte group was not known before.