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

Abstract

We consider structural properties of $\mathcal{L}$-polymatroids, especially those defined on a finite complemented modular lattice $\mathcal{L}$. We introduce a set of cover-weight axioms and establish 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 $\mathcal{P}$, along with the atomic weights of $\mathcal{P}$, is sufficient to define its rank function on $\mathcal{L}$. In our main result, we characterize those weighted lattices $(\mathcal{Z},\lambda)$ such that $\mathcal{Z}\subseteq\mathcal{L}$ is the collection of cyclic flats of an $\mathcal{L}$-polymatroid.