Independent sets of non-geometric lattices and the maximal adjoint

TL;DR

The paper develops a unified framework to study adjoint matroids through independent sets in finite, atomic, and graded lattices, extending the matroid–geometric lattice correspondence beyond geometric lattices. It introduces a robust embedding theorem into geometric lattices that preserves atoms, and uses this to derive new characterizations of adjoints, relate them to the combinatorial derived matroid, and propose algorithms to compute adjoint lists. The work establishes key results for representable and co-rank-3 matroids, identifies when derived or modified derived constructions yield adjoints, and analyzes lattice operations and their impact on independent sets. Altogether, it advances the theory of adjoints and derived matroids, offering concrete criteria, algorithmic tools, and partial proofs toward longstanding conjectures with potential computational applications.

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.