Infinite matroids

Wagstaff, Samuel S.

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

Infinite matroids
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by Samuel S. Wagstaff

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

DOI: https://doi.org/10.1090/S0002-9947-1973-0398867-7

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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, $|A|$ 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.

References

Jack Edmonds, Minimum partition of a matroid into independent subsets, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 67–72. MR 190025
Jack Edmonds and D. R. Fulkerson, Transversals and matroid partition, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 147–153. MR 188090

Ramsey’s theorem for a class of categories

Alfred Horn, A characterization of unions of linearly independent sets, J. London Math. Soc. 30 (1955), 494–496. MR 71487, DOI 10.1112/jlms/s1-30.4.494
A. W. Ingleton, A note on independence functions and rank, J. London Math. Soc. 34 (1959), 49–56. MR 101848, DOI 10.1112/jlms/s1-34.1.49
Nathan Jacobson, Lectures in abstract algebra. Vol III: Theory of fields and Galois theory, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London-New York, 1964. MR 0172871
T. Lazarson, The representation problem for independence functions, J. London Math. Soc. 33 (1958), 21–25. MR 98701, DOI 10.1112/jlms/s1-33.1.21
Alfred Lehman, A solution of the Shannon switching game, J. Soc. Indust. Appl. Math. 12 (1964), 687–725. MR 173250
Saunders MacLane, Some Interpretations of Abstract Linear Dependence in Terms of Projective Geometry, Amer. J. Math. 58 (1936), no. 1, 236–240. MR 1507146, DOI 10.2307/2371070
Saunders Mac Lane, A lattice formulation for transcendence degrees and $p$-bases, Duke Math. J. 4 (1938), no. 3, 455–468. MR 1546067, DOI 10.1215/S0012-7094-38-00438-7
R. Rado, Axiomatic treatment of rank in infinite sets, Canad. J. Math. 1 (1949), 337–343. MR 31993, DOI 10.4153/cjm-1949-031-1
R. Rado, Note on independence functions, Proc. London Math. Soc. (3) 7 (1957), 300–320. MR 88459, DOI 10.1112/plms/s3-7.1.300
R. Rado, A combinatorial theorem on vector spaces, J. London Math. Soc. 37 (1962), 351–353. MR 146186, DOI 10.1112/jlms/s1-37.1.351
W. T. Tutte, A homotopy theorem for matroids. I, II, Trans. Amer. Math. Soc. 88 (1958), 144–174. MR 101526, DOI 10.1090/S0002-9947-1958-0101526-0
W. T. Tutte, Matroids and graphs, Trans. Amer. Math. Soc. 90 (1959), 527–552. MR 101527, DOI 10.1090/S0002-9947-1959-0101527-3
W. T. Tutte, Lectures on matroids, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 1–47. MR 179781

On infinite matroids

Hassler Whitney, On the Abstract Properties of Linear Dependence, Amer. J. Math. 57 (1935), no. 3, 509–533. MR 1507091, DOI 10.2307/2371182
Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
James Ax, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239–271. MR 229613, DOI 10.2307/1970573
Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory: Combinatorial geometries, Preliminary edition, The M.I.T. Press, Cambridge, Mass.-London, 1970. MR 0290980
P. Erdös and S. Kakutani, On non-denumerable graphs, Bull. Amer. Math. Soc. 49 (1943), 457–461. MR 8136, DOI 10.1090/S0002-9904-1943-07954-2
M. J. Piff, Properties of finite character of independence spaces, Mathematika 18 (1971), 201–208. MR 306024, DOI 10.1112/S0025579300005465

A necessary and sufficient condition for a matroid to be linear