Infinite matroids
Author:
Samuel S. Wagstaff
Journal:
Trans. Amer. Math. Soc. 175 (1973), 141153
MSC:
Primary 05B35
MathSciNet review:
0398867
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Abstract: Matroids axiomatize the related notions of dimension and independence. We prove that if S is a set with k matroid structures, then S is the union of k subsets, the ith of which is independent in the ith matroid structure, iff for every (finite) subset A of S, is not larger than the sum of the dimensions of A in the k matroids. A matroid is representable if there is a dimensionpreserving imbedding of it in a vector space. A matroid is constructed which is not the union of finitely many representable matroids. It is shown that a matroid is representable iff every finite subset of it is, and that if a matroid is representable over fields of characteristic p for infinitely many primes p, then it is representable over a field of characteristic 0. Similar results for other kinds of representation are obtained.
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 H. Whitney, On the abstract properties of linear independence, Amer. J. Math. 57 (1935), 509533. MR 1507091
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 J. Ax, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239271. MR 37 #5187. MR 0229613 (37:5187)
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 H. H. Crapo and GianCarlo Rota, On the foundations of combinatorial theory: Combinatorial geometries, preliminary edition, M.I.T. Press, Cambridge, Mass., 1970. MR 0290980 (45:74)
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 P. Erdös and S. Kakutani, On nondenumerable graphs, Bull. Amer. Math. Soc. 49 (1943), 457461. MR 4, 249. MR 0008136 (4:249f)
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 P. Vámos, A necessary and sufficient condition for a matroid to be linear, Matroid Conference, Brest, 1970 (preprint).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197303988677
PII:
S 00029947(1973)03988677
Keywords:
Matroid,
dimension function,
transcendence degree,
matroid representation problem,
purely inseparable extension
Article copyright:
© Copyright 1973
American Mathematical Society
