Independent sets of non-geometric lattices and the maximal adjoint

Abstract

We construct a family of independent sets for finite, atomic, and graded lattices, extending the well-known cryptomorphism between geometric lattices and matroids. This construction leads to an embedding theorem into geometric lattices that preserves the set of atoms. We then apply these results to adjoint matroids, providing new characterizations of adjoints and partially proving a conjecture on the combinatorial derived matroid. Finally, we use our characterization of adjoints to compute the adjoint lists of several simple examples.