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Extensions of transversal valuated matroids

Alex Fink, Jorge Alberto Olarte

Abstract

Following up on our previous work, we study single-element extensions of transversal valuated matroids. We show that tropical presentations of valuated matroids with a minimal set of finite entries enjoy counterparts of the properties proved by Bonin and de Mier of minimal non-valuated transversal presentations.

Extensions of transversal valuated matroids

Abstract

Following up on our previous work, we study single-element extensions of transversal valuated matroids. We show that tropical presentations of valuated matroids with a minimal set of finite entries enjoy counterparts of the properties proved by Bonin and de Mier of minimal non-valuated transversal presentations.
Paper Structure (7 sections, 11 theorems, 16 equations)

This paper contains 7 sections, 11 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Matroids
  4. Valuated matroids
  5. Extensions
  6. Transversal (valuated) matroids and presentations
  7. Minimal transversal presentations and their extensions

Key Result

Proposition 3.1

Any transversal matroid $M$ has a unique maximal presentation. A presentation $\mathcal{A}$ of $M$ is minimal if and only if $A_i$ is a cocircuit of $M$ for every $i$.

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.2
  • Proposition 3.1: BrualdiDinolt
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Proposition 3.5: FinkOlarte
  • Definition 3.6
  • Theorem 3.7
  • proof
  • ...and 18 more