Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
axiom of choice; finitary matroid; circuit; hyperplane; graph
axiom of choice; finitary matroid; circuit; hyperplane; graph
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?
References:
[1] Cohn P. M.: Universal Algebra. Mathematics and Its Applications, 6, D. Reidel Publishing Co., Dordrecht, 1981. MR 0620952 | Zbl 0461.08001
[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
[3] Higgs D. A.: Matroids and duality. Colloq. Math. 20 (1969), 215–220. DOI 10.4064/cm-20-2-215-220 | MR 0274315
[4] Hodges W.: Krull implies Zorn. J. London Math. Soc. (2) 19 (1979), no. 2, 285–287. MR 0533327
[5] Höft H., Howard P.: A graph theoretic equivalent to the axiom of choice. Z. Math. Logik Grundlagen Math. 19 (1973), 191. DOI 10.1002/malq.19730191103 | MR 0316283
[6] Howard P.: Bases, spanning sets, and the axiom of choice. MLQ Math. Log. Q. 53 (2007), no. 3, 247–254. MR 2330594 | Zbl 1121.03064
[7] Howard P., Rubin J. E.: Consequences of the Axiom of Choice. Mathematical Surveys and Monographs, 59, American Mathematical Society, Providence, 1998. DOI 10.1090/surv/059 | MR 1637107 | Zbl 0947.03001
[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. MR 0396271 | Zbl 0259.02052
[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
[10] Morillon M.: Linear forms and axioms of choice. Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421–431. MR 2573415 | Zbl 1212.03034
[11] Morillon M.: Multiple choices imply the Ingleton and Krein–Milman axioms. J. Symb. Log. 85 (2020), no. 1, 439–455. DOI 10.1017/jsl.2019.48 | MR 4085068
[12] Nicoletti G., White N.: Axiom Systems. Theory of Matroids. Encyclopedia Math. Appl., 26, Cambridge Univ. Press, Cambridge, 1986, pages 29–44. MR 0849391
[13] Oxley J. G.: Infinite matroids. Proc. London Math. Soc. (3) 37 (1978), no. 2, 259–272. MR 0507607
[14] Oxley J.: Matroid Theory. Oxford Graduate Texts in Mathematics, 21, Oxford University Press, Oxford, 2011. MR 2849819
[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
[16] Welsh D. J. A.: Matroid Theory. L. M. S. Monographs, 8, Academic Press, London, 1976. MR 0427112 | Zbl 0343.05002
[17] Zariski O., Samuel P.: Commutative Algebra. Vol. 1. Graduate Texts in Mathematics, 28, Springer, New York, 1975. MR 0384768
Search
Partner of
