Alfter, Marion and Hochstättler, Winfried (1995) On Pseudomodular Matroids and Adjoints.
Published in: Discrete applied mathematics Vol. 60 (1-3). pp. 3-11.

Abstract

There are two concepts of duality in combinatorial geometry. A set theoretical one, generalizing the structure of two orthocomplementary vector spaces and a lattice theoretical concept of an adjoint, that mimics duality between points and hyperplanes. The latter - usually called polarity - seems to make sense almost only in the linear case. In fact the only non-linear combinatorial geometries known to admit an adjoint were of rank 3. Moreover, N. E. Mnëv conjectured that in higher ranks there would exist no non-linear oriented matroid that has an oriented adjoint. At least with unoriented matroids this is not true. In this paper we present a class of rank 4 matroids with adjoint including a non-linear example.

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