Decompositions of q-Matroids Using Cyclic Flats

We introduce the notion of cyclic flats for q-matroids. By expressing the rank function of the q-matroids in terms of the cyclic flats and their rank values, we obtain that the cyclic flats together with their rank values fully determine the q-matroids. Next we show that the cyclic flats of the direct sum of two q-matroids are exactly all the direct sums of the cyclic flats of the two summands. This simplifies the rank function of the direct sum significantly. A q-matroid is called irreducible if it cannot be written as a (non-trivial) direct sum. We provide a characterization of irreducibility in terms of the cyclic flats and show that every q-matroid can be decomposed into a direct sum of irreducible ones, which are unique up to equivalence.