On Rank Functions of Lattices

Abstract

Let $\mathcal{D}$ be a lattice of finite height. The correspondence between closure operators $cl:\mathcal{D} \to \mathcal{D}$ and ∧-subsemilattices $\mathcal{L} \subseteq \mathcal{D}$ is well known. Here we investigate what type of number-valued function $\mathcal{D} \to \mathbb{N}$ is induces a ∧-subsemilattice $\mathcal{L}$; and if so, what kind of $\mathcal{L}$. Conversely, what type of function $\mathcal{D} \to \mathbb{N}$ is induced by what type of $\mathcal{L}$ (or cl). Several results known for matroids, greedoids, or semimodular lattices are generalized.