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arXiv · 2603.13550

A cyclic flat embedding theorem for transversal $q$-matroids

Andrew Fulcher
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Abstract

Cyclic flats form a common structural invariant of both matroids and $q$-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 $q$-matroids that we call coordinate $q$-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 $q$-matroid setting. As an application, we show that nested $q$-matroids are transversal and therefore representable. Finally, we illustrate the usefulness of this perspective by analysing transversal $q$-matroids under binary operations. We prove that the class of transversal $q$-matroids is closed under the free product and propose a natural presentation for the direct sum motivated by the corresponding construction for matroids.

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Andrew Fulcher. 2026-03-13. A cyclic flat embedding theorem for transversal $q$-matroids. https://arxiv.org/abs/2603.13550

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