The Lattice of Cyclic Flats of a Matroid

Abstract

A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice Z of sets and a function r on Z to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to Z. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids.

Figures (1)

• Figure 1: Two matroids that have the same lattice of cyclic flats.

Theorems & Definitions (35)

• Theorem 2.1
• proof
• Lemma 2.2
• Theorem 2.3
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• proof : Proof of (\ref{['thm:axioms']}.1).