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

TL;DR

This paper establishes a cyclic flat embedding theorem linking transversal matroids and a subclass of $q$-matroids called coordinate $q$-matroids, enabling transfer of structural properties and analysis of transversal $q$-matroids.

Contribution

It introduces a cyclic flat embedding theorem for transversal $q$-matroids, connecting matroid theory to $q$-matroids and analyzing their structural properties.

Findings

Cyclic flat structure of transversal matroids is preserved in coordinate $q$-matroids.

Nested $q$-matroids are transversal and representable.

Transversal $q$-matroids are closed under free product and direct sum operations.

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.