# Cyclic Flats of Binary Matroids

## TL;DR

This paper explores the structure of cyclic flats in binary matroids, providing characterizations and mappings that deepen understanding of their lattice structures and representation over finite fields.

## Contribution

It introduces new characterizations of the lattice of cyclic flats for binary matroids and describes maps relating these lattices to minors.

## Key findings

- Lattices of cyclic flats characterize binary matroids.
- Lattice of cyclic flats of simple binary matroids without isthmuses is atomic.
- Maps from cyclic flats to minors' cyclic flats are established.

## Abstract

In this paper, first steps are taken towards characterising lattices of cyclic flats $\mathcal{Z}(M)$ that belong to matroids $M$ that can be represented over a prescribed finite field $\mathbb{F}_q$. Two natural maps from $\mathcal{Z}(M)$ to the lattice of cyclic flats of a minor of $M$ are given. Binary matroids are characterised via their lattice of cyclic flats. It is shown that the lattice of cyclic flats of a simple binary matroid without isthmuses is atomic.