Article
| Title: | Hyperplanes in matroids and the axiom of choice (English) |
| Author: | Morillon, Marianne |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 63 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 423-441 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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? (English) |
| Keyword: | axiom of choice |
| Keyword: | finitary matroid |
| Keyword: | circuit |
| Keyword: | hyperplane |
| Keyword: | graph |
| MSC: | 03E25 |
| MSC: | 05B99 |
| idZBL: | Zbl 07729552 |
| idMR: | MR4577876 |
| DOI: | 10.14712/1213-7243.2023.010 |
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| Date available: | 2023-04-20T13:47:50Z |
| Last updated: | 2023-10-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151644 |
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| Reference: | [1] Cohn P. M.: Universal Algebra.Mathematics and Its Applications, 6, D. Reidel Publishing Co., Dordrecht, 1981. Zbl 0461.08001, MR 0620952 |
| Reference: | [2] Fournier J.-C.: Introduction à la notion de matroïde.Géométrie combinatoire, Mathematical Publications of Orsay 79, 3, Université de Paris-Sud, Département de Mathématique, Orsay, 1979, pages 57 (French). MR 0551494 |
| Reference: | [3] Higgs D. A.: Matroids and duality.Colloq. Math. 20 (1969), 215–220. MR 0274315, 10.4064/cm-20-2-215-220 |
| Reference: | [4] Hodges W.: Krull implies Zorn.J. London Math. Soc. (2) 19 (1979), no. 2, 285–287. MR 0533327 |
| Reference: | [5] Höft H., Howard P.: A graph theoretic equivalent to the axiom of choice.Z. Math. Logik Grundlagen Math. 19 (1973), 191. MR 0316283, 10.1002/malq.19730191103 |
| Reference: | [6] Howard P.: Bases, spanning sets, and the axiom of choice.MLQ Math. Log. Q. 53 (2007), no. 3, 247–254. Zbl 1121.03064, MR 2330594 |
| Reference: | [7] Howard P., Rubin J. E.: Consequences of the Axiom of Choice.Mathematical Surveys and Monographs, 59, American Mathematical Society, Providence, 1998. Zbl 0947.03001, MR 1637107, 10.1090/surv/059 |
| Reference: | [8] Jech T. J.: The Axiom of Choice.Studies in Logic and the Foundations of Mathematics, 75, North-Holland Publishing Co., Amsterdam, American Elsevier Publishing Co., New York, 1973. Zbl 0259.02052, MR 0396271 |
| Reference: | [9] Klee V.: The greedy algorithm for finitary and cofinitary matroids.Combinatorics, Proc. Symp. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif. 1968, Amer. Math. Soc., Providence, 1971, pages 137–152. MR 0332538 |
| Reference: | [10] Morillon M.: Linear forms and axioms of choice.Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421–431. Zbl 1212.03034, MR 2573415 |
| Reference: | [11] Morillon M.: Multiple choices imply the Ingleton and Krein–Milman axioms.J. Symb. Log. 85 (2020), no. 1, 439–455. MR 4085068, 10.1017/jsl.2019.48 |
| Reference: | [12] Nicoletti G., White N.: Axiom Systems. Theory of Matroids.Encyclopedia Math. Appl., 26, Cambridge Univ. Press, Cambridge, 1986, pages 29–44. MR 0849391 |
| Reference: | [13] Oxley J. G.: Infinite matroids.Proc. London Math. Soc. (3) 37 (1978), no. 2, 259–272. MR 0507607 |
| Reference: | [14] Oxley J.: Matroid Theory.Oxford Graduate Texts in Mathematics, 21, Oxford University Press, Oxford, 2011. MR 2849819 |
| Reference: | [15] Rubin H., Rubin J. E.: Equivalents of the Axiom of Choice.Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1970. MR 0434812 |
| Reference: | [16] Welsh D. J. A.: Matroid Theory.L. M. S. Monographs, 8, Academic Press, London, 1976. Zbl 0343.05002, MR 0427112 |
| Reference: | [17] Zariski O., Samuel P.: Commutative Algebra. Vol. 1.Graduate Texts in Mathematics, 28, Springer, New York, 1975. MR 0384768 |
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