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.

Abstract

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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