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Journal Articles Commentationes Mathematicae Universitatis Carolinae Year : 2022

Hyperplanes in matroids and the axiom of choice

Abstract

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 fin , the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC fin in terms of “graphic” matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?
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Dates and versions

hal-04524324 , version 1 (28-03-2024)

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Marianne Morillon. Hyperplanes in matroids and the axiom of choice. Commentationes Mathematicae Universitatis Carolinae, 2022, 63 (4), pp.423-441. ⟨10.14712/1213-7243.2023.010⟩. ⟨hal-04524324⟩
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