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 cyclic flats, together with their ranks, uniquely determine a q-matroid and hence derive a new q-cryptomorphism. We introduce the notion of $$\mathbb {F}_{q^m}$$ F q m -independence of an $$\mathbb {F}_q$$ F q -subspace of $$\mathbb {F}_q^n$$ 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.