Decompositions of \(q\)-Matroids Using Cyclic Flats

We study the direct sum of \(q\)-matroids by way of their cyclic flats. Using that the rank function of a \(q\)-matroid is fully determined by the cyclic flats and their ranks, 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 (nontrivial) 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 \(q\)-matroids, which are unique up to equivalence.<p/>