On pseudomodular matroids and adjoints Original Research Article

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.