On Adjoints and Dual Matroids

Alfter, Marion and Kern, Walter and Wanka, Alfred (1990) On Adjoints and Dual Matroids.
Published in: Journal of combinatorial theory : Series B Vol. 50 (2). pp. 208-213.


Duality among matroids is a well-known and well-understood relation. Besides this ''set-theoretical'' version of duality, there is another one based on lattice-theoretical concepts, which has been introduced by A. Cheung [Adjoints of a geometry, Canadian Math. Bul. 17, 363-365 (1974)]. These two concepts do not seem to fit into one another very well and their relationship (provided there is any) is more than unclear. In general, matroids may fail to have ''duals'' in the lattice-theoretical sense. Therefore, a natural question, posed by J.H. Mason [Glueing matroids together: a study of Dilworth truncations and matroid analogues of exterior and symmetric powers, in: Algebraic Methods in graph theory, L. Lovász and V.T. Sôs, eds., North-Holland, Amsterdam (1981), is the following: If M does have a dual in the lattice theoretical sense, does M * (the set-theoretical dual of M) also have one? We present a counterexample, showing that the answer is negative.

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Item Type: Article
Citations: 9 (Google Scholar) | 0 (Web of Science)
Uncontrolled Keywords: dual matroid
  • 05-XX Combinatorics > 05Bxx Designs and configurations > 05B35 Matroids, geometric lattices

  • Divisions: Mathematical Institute
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    ZAIK Number: zpr88-049
    Depositing User: Archive Admin
    Date Deposited: 02 Apr 2001 00:00
    Last Modified: 09 Jan 2012 11:51
    URI: http://e-archive.informatik.uni-koeln.de/id/eprint/49