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 $\mathbin{\square}$-irreducible with respect to the free product and we prove that the factorization of a $q$-matroid into a free product of $\mathbin{\square}$-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.
@article{10_37236_13647,
author = {Gianira N. Alfarano and Eimear Byrne and Andrew Fulcher},
title = {The free product of \(q\)-matroids},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {3},
doi = {10.37236/13647},
zbl = {8097680},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13647/}
}
TY - JOUR
AU - Gianira N. Alfarano
AU - Eimear Byrne
AU - Andrew Fulcher
TI - The free product of \(q\)-matroids
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/13647/
DO - 10.37236/13647
ID - 10_37236_13647
ER -
%0 Journal Article
%A Gianira N. Alfarano
%A Eimear Byrne
%A Andrew Fulcher
%T The free product of \(q\)-matroids
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/13647/
%R 10.37236/13647
%F 10_37236_13647
Gianira N. Alfarano; Eimear Byrne; Andrew Fulcher. The free product of \(q\)-matroids. The electronic journal of combinatorics, Tome 32 (2025) no. 3. doi: 10.37236/13647
