Hyperplanes in matroids and the axiom of choice

[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.

[学科分类] 物理化学和理论化学

[关键词] axiom of choice;finitary matroid;circuit;hyperplane;graph