The Cyclic Flats of a $q$-Matroid

TL;DR

This paper develops the theory of cyclic flats in $q$-matroids, showing they uniquely determine the structure and introducing a new cryptomorphism, thereby generalizing linear independence concepts over finite fields.

Contribution

It introduces the lattice of cyclic flats for $q$-matroids, establishing a new cryptomorphism and linking $q$-matroids to $F_{q^m}$-independence.

Findings

Cyclic flats and their ranks uniquely determine a $q$-matroid.

A new $q$-cryptomorphism is derived.

$q$-matroids generalize $F_{q^m}$-independence.

Abstract

In this paper we develop the theory of cyclic flats of $q$-matroids. We show that the lattice of cyclic flats, together with their ranks, uniquely determines a $q$-matroid and hence derive a new $q$-cryptomorphism. We introduce the notion of $\mathbb{F}_{q^m}$-independence of an $\mathbb{F}_q$-subspace of $\mathbb{F}_q^n$ and we show that $q$-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.