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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Self-dual uniform matroids on infinite sets
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by Nathan Bowler and Stefan Geschke PDF
Proc. Amer. Math. Soc. 144 (2016), 459-471 Request permission

Abstract:

We extend the notion of a uniform matroid to the infinitary case and construct, using weak fragments of Martin’s Axiom, self-dual uniform matroids on infinite sets. In 1969, Higgs showed that, assuming the Generalised Continuum Hypothesis (GCH), any two bases of a fixed matroid have the same size. We show that this cannot be proved from the usual axioms of set theory, ZFC, alone: in fact, we show that it is consistent with ZFC that there is a uniform self-dual matroid with two bases of different sizes.

Self-dual uniform matroids on infinite sets also provide examples of infinitely connected matroids, answering a question of Bruhn and Wollan under additional set-theoretic assumptions. While we do not know whether the existence of a self-dual uniform matroid on an infinite set can be proved in ZFC alone, we show that ZF, Zermelo-Fraenkel Set Theory without the Axiom of Choice, is not enough. Finally, we observe that there is a model of set theory in which GCH fails while any two bases of a matroid have the same size. This answers a question of Higgs.

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Additional Information
  • Nathan Bowler
  • Affiliation: Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146 Hamburg, Germany
  • Email: N.Bowler1729@gmail.com
  • Stefan Geschke
  • Affiliation: Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146 Hamburg, Germany
  • MR Author ID: 681801
  • Email: stefan.geschke@uni-hamburg.de
  • Received by editor(s): February 28, 2014
  • Received by editor(s) in revised form: October 6, 2014
  • Published electronically: October 7, 2015
  • Additional Notes: We thank Johannes Carmesin for many fruitful discussions on the subject of this article.
  • Communicated by: Mirna Džamonja
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 459-471
  • MSC (2010): Primary 05B35; Secondary 03E35, 03E25, 03E50
  • DOI: https://doi.org/10.1090/proc/12667
  • MathSciNet review: 3430826