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Yes, the “missing axiom” of matroid theory is lost forever. (English) Zbl 06871590
Summary: We prove there is no sentence in the monadic second-order language $$MS_0$$ that characterises when a matroid is representable over at least one field, and no sentence that characterises when a matroid is $$\mathbb{K}$$-representable, for any infinite field $$\mathbb{K}$$. By way of contrast, because Rota’s Conjecture is true, there is a sentence that characterises $$\mathbb{F}$$-representable matroids, for any finite field $$\mathbb{F}$$.

##### MSC:
 03C13 Model theory of finite structures 05B35 Combinatorial aspects of matroids and geometric lattices
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##### References:
 [1] Aigner, Martin, Combinatorial theory, reprint of the 1979 original, Classics in Mathematics, viii+483 pp., (1997), Springer-Verlag, Berlin · Zbl 0858.05001 [2] Courcelle, Bruno, The monadic second-order logic of graphs. I. Recognizable sets of finite graphs, Inform. and Comput., 85, 1, 12-75, (1990) · Zbl 0722.03008 [3] Ebbinghaus, Heinz-Dieter; Flum, J\"org, Finite model theory, Springer Monographs in Mathematics, xii+360 pp., (2006), Springer-Verlag, Berlin · Zbl 1081.03026 [4] Geelen, Jim, Some open problems on excluding a uniform matroid, Adv. in Appl. Math., 41, 4, 628-637, (2008) · Zbl 1172.05013 [5] Geelen, Jim; Gerards, Bert; Whittle, Geoff, Solving Rota’s conjecture, Notices Amer. Math. Soc., 61, 7, 736-743, (2014) · Zbl 1338.05039 [6] Hlin\v en\'y, Petr, On matroid properties definable in the MSO logic. Mathematical foundations of computer science 2003, Lecture Notes in Comput. Sci. 2747, 470-479, (2003), Springer, Berlin · Zbl 1124.68373 [7] Lengauer, Thomas; Wanke, Egon, Efficient analysis of graph properties on context-free graph languages (extended abstract). Automata, languages and programming, Tampere, 1988, Lecture Notes in Comput. Sci. 317, 379-393, (1988), Springer, Berlin · Zbl 0649.68076 [8] Libkin, Leonid, Elements of finite model theory, Texts in Theoretical Computer Science. An EATCS Series, xiv+315 pp., (2004), Springer-Verlag, Berlin · Zbl 1060.03002 [9] Mayhew, Dillon; Whittle, Geoff; Newman, Mike, Is the missing axiom of matroid theory lost forever?, Q. J. Math., 65, 4, 1397-1415, (2014) · Zbl 1305.05040 [10] Nerode, A., Linear automaton transformations, Proc. Amer. Math. Soc., 9, 541-544, (1958) · Zbl 0089.33403 [11] Oxley, James, Matroid theory, Oxford Graduate Texts in Mathematics 21, xiv+684 pp., (2011), Oxford University Press, Oxford · Zbl 1254.05002 [12] V\'amos, P., The missing axiom of matroid theory is lost forever, J. London Math. Soc. (2), 18, 3, 403-408, (1978) · Zbl 0395.05024 [13] Whitney, Hassler, On the abstract properties of linear dependence, Amer. J. Math., 57, 3, 509-533, (1935) · Zbl 0012.00404 [14] Zaslavsky, Thomas, Biased graphs. IV. Geometrical realizations, J. Combin. Theory Ser. B, 89, 2, 231-297, (2003) · Zbl 1031.05034
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