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Equicardinality of Bases in B-Matroids

Published online by Cambridge University Press:  20 November 2018

Denis Higgs
Denis Higgs*
Affiliation:
University of Waterloo
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It is very well known that any two bases of a finitary matroid (see [2] for definitions) have the same cardinality. As Dlab has shown in [1], the same does not hold for arbitrary transitive exchange spaces; indeed, since the examples Dlab constructs in [1] are matroids, it does not even hold for arbitrary matroids. Nevertheless with the aid of the generalized continuum hypothesis (G. C.H.) we are able to prove the result for B-matroids.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 6 , December 1969 , pp. 861 - 862
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Dlab, V., The role of the "finite character property" in the theory of dependence. Comment. Math. Univ. Carolinae 6 (1965) 97–104.Google Scholar
2. Higgs, D.A., Matroids and duality. Colloq. Math. 20 (1969) 215–220.Google Scholar
3. Sierpinski, W., Sur un problème concernant les sous-ensembles croissant du continu. Fund. Math. 3 (1922) 109–112.Google Scholar
4. Wolk, E.S., A theorem on power sets. Amer. Math. Monthly 72 (1965) 397–398.Google Scholar