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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Infinite matroids


Author: Samuel S. Wagstaff
Journal: Trans. Amer. Math. Soc. 175 (1973), 141-153
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, $ \vert A\vert$ is not larger than the sum of the dimensions of A in the k matroids.

A matroid is representable if there is a dimension-preserving 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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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0398867-7
PII: S 0002-9947(1973)0398867-7
Keywords: Matroid, dimension function, transcendence degree, matroid representation problem, purely inseparable extension
Article copyright: © Copyright 1973 American Mathematical Society