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

Andrew Fulcher

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.

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.
Paper Structure (4 sections, 19 theorems, 35 equations)

This paper contains 4 sections, 19 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The cyclic flat embedding theorem
  4. Closure of q-transversal matroids under binary operations

Key Result

Theorem 2.9

GluesingLuerssenJany2024 For $q$-matroids $M_1$ and $M_2$,

Theorems & Definitions (47)

  • Definition 2.2
  • Definition 2.4
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • Definition 2.13
  • ...and 37 more