A cyclic flat embedding theorem for transversal $q$-matroids
Abstract
Cyclic flats form a common structural invariant of both matroids and -matroids, determining these objects through their weighted lattices of cyclic flats. In this paper we exploit this perspective to establish a correspondence between matroids and a subclass of -matroids that we call coordinate -matroids. Our main result is a cyclic flat embedding theorem showing that the cyclic flat structure of a transversal matroid is preserved under this correspondence. This provides a mechanism for transferring structural properties from matroid theory to the -matroid setting. As an application, we show that nested -matroids are transversal and therefore representable. Finally, we illustrate the usefulness of this perspective by analysing transversal -matroids under binary operations. We prove that the class of transversal -matroids is closed under the free product and propose a natural presentation for the direct sum motivated by the corresponding construction for matroids.
Cite
@article{arxiv.2603.13550,
title = {A cyclic flat embedding theorem for transversal $q$-matroids},
author = {Andrew Fulcher},
journal= {arXiv preprint arXiv:2603.13550},
year = {2026}
}