We introduce the notion of the free product of $q$-matroids, which is the $q$-analogue of the free product of matroids. We study the properties of this noncommutative binary operation, making an extensive use of the theory of cyclic flats. We show that the free product of two $q$-matroids $M_1$ and $M_2$ is maximal with respect to the weak order on $q$-matroids having $M_1$ as a restriction and $M_2$ as the complementary contraction. We characterise $q$-matroids that are irreducible with respect to the free product and we prove that the factorization of a $q$-matroid into a free product of irreducibles is unique up to isomorphism. We discuss the representability of the free product, with a particular focus on rank one uniform $q$-matroids and show that such a product is represented by clubs on the projective line.

翻译：我们引入了q-拟阵自由积的概念，这是拟阵自由积的q-模拟。我们利用循环平坦理论深入研究了这一非交换二元运算的性质。我们证明，对于两个q-拟阵M₁和M₂，其自由积在弱序意义下是最大的，且满足以M₁为限制、M₂为互补收缩的条件。我们刻画了关于自由积不可约的q-拟阵，并证明了将q-拟阵分解为不可约自由积的表示在同构意义下是唯一的。我们讨论了自由积的可表示性，特别聚焦于秩为一的均匀q-拟阵，并证明此类积可由射影直线上的club结构表示。