## 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

*i*th of which is independent in the

*i*th 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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## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**175**(1973), 141-153 - MSC: Primary 05B35
- DOI: https://doi.org/10.1090/S0002-9947-1973-0398867-7
- MathSciNet review: 0398867