Infinite matroids

Wagstaff, Samuel S.

doi:10.1090/S0002-9947-1973-0398867-7

Infinite matroids
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by Samuel S. Wagstaff PDF

Trans. Amer. Math. Soc. 175 (1973), 141-153 Request permission

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, $|A|$ 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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