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arXiv · 2312.05522

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

Eimear Byrne·Andrew Fulcher
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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 $(\mathcal{Z},λ)$ such that $\mathcal{Z}\subseteq\mathcal{L}$ is the collection of cyclic flats of an $\mathcal{L}$-polymatroid.

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Eimear Byrne, Andrew Fulcher. 2025-12-28. The cyclic flats of $\mathcal{L}$-polymatroids. https://arxiv.org/abs/2312.05522

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