Hyperplanes in matroids and the axiom of choice

Commentationes mathematicae Universitatis Carolinae

article

Marianne Morillon

关键词: axiom of choice;    finitary matroid;    circuit;    hyperplane;    graph;   

DOI  :  10.14712/1213-7243.2023.010

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

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.

【 授权许可】

CC BY