{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-counterexample-to-las-vergnas-strong-map","title":"A counterexample to Las Vergnas' strong map conjecture on realizable oriented matroids","arxiv_id":"1803.06825","date":"2018-03-19","proceeding":null,"authors":["Pei Wu"],"abstract":"The Las Vergnas' strong map conjecture, states that any strong map of oriented matroids $f:\\mathcal{M}_1\\rightarrow\\mathcal{M}_2$ can be factored into extensions and contractions. The conjecture is known to be false due to a construction by Richter-Gebert, he find a non-factorizable strong map $f:\\mathcal{M}_1\\rightarrow\\mathcal{M}_2$, however in his example $\\mathcal{M}_1$ is not realizable. The problem that whether there exists a non-factorizable strong map between realizable oriented matroids still remains open. In this paper we provide a counterexample to the strong map conjecture on realizable oriented matroids, which is a strong map $f:\\mathcal{M}_1\\rightarrow\\mathcal{M}_2$, $\\mathcal{M}_1$ is an alternating oriented matroid of rank $4$ and $f$ has corank $2$. We prove it is not factorizable by showing that there is no uniform oriented matroid $\\mathcal{M}^{\\prime}$ of rank $3$ such that $\\mathcal{M}_1\\rightarrow\\mathcal{M}^{\\prime}\\rightarrow\\mathcal{M}_2$.","url_abs":"http://arxiv.org/abs/1803.06825v2","url_pdf":"http://arxiv.org/pdf/1803.06825v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-counterexample-to-las-vergnas-strong-map","repo_url":"https://github.com/PeterWu-Biomath/OM-Stong-Map","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}