Universal partial hyperfields of matroids and their prespaces of orderings
| dc.contributor.advisor | Kalhoff, Franz | |
| dc.contributor.author | Hoya, Marcel | |
| dc.contributor.referee | Wenzel, Walter | |
| dc.date.accepted | 2023-04-13 | |
| dc.date.accessioned | 2024-04-03T08:29:54Z | |
| dc.date.available | 2024-04-03T08:29:54Z | |
| dc.date.issued | 2023 | |
| dc.description.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. | en |
| dc.identifier.uri | http://hdl.handle.net/2003/42414 | |
| dc.identifier.uri | http://dx.doi.org/10.17877/DE290R-24250 | |
| dc.language.iso | en | de |
| dc.subject | Partial hyperfields | en |
| dc.subject | Inner Tutte group | en |
| dc.subject | Matroids | en |
| dc.subject | Combinatorial geometries | en |
| dc.subject | Affine planes | en |
| dc.subject | Affine spaces | en |
| dc.subject | Projective planes | en |
| dc.subject | Projective spaces | en |
| dc.subject.ddc | 510 | |
| dc.subject.rswk | Orientiertes Matroid | de |
| dc.subject.rswk | Kombinatorische Geometrie | de |
| dc.subject.rswk | Körpertheorie | de |
| dc.title | Universal partial hyperfields of matroids and their prespaces of orderings | en |
| dc.type | Text | de |
| dc.type.publicationtype | PhDThesis | de |
| dcterms.accessRights | open access | |
| eldorado.secondarypublication | false | de |