Kalhoff, FranzHoya, Marcel2024-04-032024-04-032023http://hdl.handle.net/2003/42414http://dx.doi.org/10.17877/DE290R-24250We 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.enPartial hyperfieldsInner Tutte groupMatroidsCombinatorial geometriesAffine planesAffine spacesProjective planesProjective spaces510TextOrientiertes MatroidKombinatorische GeometrieKörpertheorie