The Cyclic Flats of a $q$-Matroid

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.