The Lattice of Cyclic Flats of a Matroid

TL;DR

This paper investigates the structure of cyclic flats in matroids, proving that any lattice can be realized as the lattice of cyclic flats of some matroid and characterizing these lattices.

Contribution

It provides a necessary and sufficient condition for a lattice and rank function to correspond to the cyclic flats of a matroid, introducing the concept of cyclic width.

Findings

Every lattice is isomorphic to the lattice of cyclic flats of a matroid.

A characterization criterion for lattices of cyclic flats is established.

Cyclic width leads to minor-closed, dual-closed classes of matroids, including transversal matroids.

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.