Independent sets of non-geometric lattices and the maximal adjoint

TL;DR

This paper introduces a new construction of independent sets in finite lattices, extends the cryptomorphism between geometric lattices and matroids, and applies these results to characterize and compute adjoint matroids.

Contribution

It develops a novel method for constructing independent sets in lattices, leading to new insights into adjoint matroids and their properties.

Findings

Extended cryptomorphism between geometric lattices and matroids

Provided new characterizations of adjoint matroids

Computed adjoint lists for several examples

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.