Hyperplanes in matroids and the axiom of choice

Marianne Morillon

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 4, page 423-441
  • ISSN: 0010-2628

Abstract

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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 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?

How to cite

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Morillon, Marianne. "Hyperplanes in matroids and the axiom of choice." Commentationes Mathematicae Universitatis Carolinae 62 63.4 (2022): 423-441. <http://eudml.org/doc/299040>.

@article{Morillon2022,
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$^\{\rm fin\}$, 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?},
author = {Morillon, Marianne},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {axiom of choice; finitary matroid; circuit; hyperplane; graph},
language = {eng},
number = {4},
pages = {423-441},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Hyperplanes in matroids and the axiom of choice},
url = {http://eudml.org/doc/299040},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Morillon, Marianne
TI - Hyperplanes in matroids and the axiom of choice
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 4
SP - 423
EP - 441
AB - 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?
LA - eng
KW - axiom of choice; finitary matroid; circuit; hyperplane; graph
UR - http://eudml.org/doc/299040
ER -

References

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