Geodesic
Hyperplanes in matroids and the axiom of choice
Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 4, pp. 423-441.

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We show that in set theory without the axiom of choice ZF, the statement sH: ``Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it'' implies AC$^{\rm fin}$, the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC$^{\rm fin}$ in terms of ``graphic'' matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?
DOI : 10.14712/1213-7243.2023.010
Classification : 03E25, 05B99
Keywords: axiom of choice; finitary matroid; circuit; hyperplane; graph 
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Morillon, Marianne. Hyperplanes in matroids and the axiom of choice. Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 4, pp. 423-441. doi : 10.14712/1213-7243.2023.010. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2023.010/

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