Independent sets of non-geometric lattices and the maximal adjoint

We present a construction of a family of independent sets for a finite, atomic and graded lattice generalizing the known cryptomorphism between geometric lattices and matroids. This family then gives rise to an embedding theorem into geometric lattices preserving the set of atoms. Lastly we apply these theorems to the concept of adjoint matroids obtaining a characterization and proving a conjecture on the combinatorial derived madtroid for uniform and co-rank 3 matroids

On pseudomodular matroids and adjoints

AbstractThere 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

The Category of Matroids

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial.Comment: 31 pages, 10 diagrams, 28 reference

A Note on Extension Properties and Representations of Matroids

We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic representations. Iterations of those extension properties are checked for matroids on eight and nine elements by means of computer-aided explorations, finding in that way several new examples of non-linearly representable matroids. A special emphasis is made on sparse paving matroids on nine points containing the tic-tac-toe configuration. We present a clear description of that family and we analyze extension properties on those matroids and their duals