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

TL;DR

The paper generalizes cyclic-flat theory to $\\mathcal{L}$-polymatroids on finite complemented modular lattices, introducing cover-weight axioms and establishing cryptomorphisms with rank functions. It defines cyclic flats and shows that the rank on the ambient lattice is recoverable from the weighted lattice of cyclic flats together with atom weights, via $r(X)=\\min\\{ r(Z)+\\mu_r(Z^c\\wedge X): Z\\in\\mathcal{Z}, Z^c\\in \\mathbf{C}(Z;X)\\}$. The core contribution is a six-axiom characterization that precisely describes when a weighted lattice $(\\mathcal{Z},\\lambda)$ embeds as the cyclic-flat lattice of some $\\mathcal{L}$-polymatroid, thereby extending cryptomorphisms known for matroids and $q$-matroids to this broader setting. This framework enables compact representations and structural analysis of polymatroids on complex lattice types and connects lattice-theoretic properties with rank-convolution constructions. The results have potential implications for generalized matroid-like structures in combinatorics and coding theory where the ambient lattice is a product of Boolean and projective-geometric components.

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.

Figures (8)

• Figure 1: A $q$-polymatroid on $\mathbb{F}_2^3$ with cover-weights (0,1,2).
• Figure 2: A $q$-polymatroid on $\mathbb{F}_2^3$ with cover-weights (0,1,2).
• Figure 3: A $q$-polymatroid with cover weights (0,1,2).
• Figure 4: On the left is the weighted lattice $(\mathcal{Z},\lambda)$. On the right is the lattice $\mathcal{L}$, whose atoms are weighted by $f$. The triple $(\mathcal{Z},\lambda,f)$ satisfies \ref{['z1']}--\ref{['z6']}. (1, 2, 3, 4)
• Figure 5: A representation of the $\mathcal{L}$-polymatroid $(\mathcal{L},\rho_{(\lambda,f)})$ whose lattice of cyclic flats is $(\mathcal{Z},\lambda,f)$, as shown in Figure \ref{['fig:cyc_flat_example']}. (1, 2, 3, 4)

Theorems & Definitions (107)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3: Isomorphism Theorem for Modular Lattices
• Definition 2.4
• Lemma 2.5
• Lemma 2.6
• proof
• Corollary 2.7
• Definition 2.8
• Proposition 2.9