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Transactions of the American Mathematical Society

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Yes, the ``missing axiom'' of matroid theory is lost forever


Authors: Dillon Mayhew, Mike Newman and Geoff Whittle
Journal: Trans. Amer. Math. Soc. 370 (2018), 5907-5929
MSC (2010): Primary 03C13, 05B35
DOI: https://doi.org/10.1090/tran/7408
Published electronically: April 17, 2018
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Abstract: We prove there is no sentence in the monadic second-order language \ensuremath {\mathit {MS}_{0}} that characterises when a matroid is representable over at least one field, and no sentence that characterises when a matroid is \ensuremath {\mathbb{K}}-representable, for any infinite field \ensuremath {\mathbb{K}}. By way of contrast, because Rota's Conjecture is true, there is a sentence that characterises \ensuremath {\mathbb{F}}-representable matroids, for any finite field \ensuremath {\mathbb{F}}.


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Additional Information

Dillon Mayhew
Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
Email: dillon.mayhew@vuw.ac.nz

Mike Newman
Affiliation: Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
Email: mnewman@uottawa.ca

Geoff Whittle
Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
Email: geoff.whittle@vuw.ac.nz

DOI: https://doi.org/10.1090/tran/7408
Received by editor(s): January 11, 2015
Received by editor(s) in revised form: March 1, 2017
Published electronically: April 17, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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