The cyclic flats of $\mathcal{L}$-polymatroids

Abstract: We consider properties of the class of $\mathcal{L}$-polymatroids, especially those that are defined on a finite modular complemented lattice $\mathcal{L}$. We give a set of cover-weight axioms and hence give a cryptomorphism between these axioms and the rank axioms of an $\mathcal{L}$-polymatroid. We introduce the notion of a cyclic flat of an $\mathcal{L}$-polymatroid and study properties of its lattice of cyclic flats. We show that the weighted lattice of cyclic flats of an $\mathcal{L}$-polymatroid along with its atomic weights are sufficient to define its rank function on $\mathcal{L}$. In our main result, we characterise those lattices $\mathcal{Z} \subseteq \mathcal{L}$ such that $\mathcal{Z}$ is the collection of cyclic flats of an $\mathcal{L}$-polymatroid.