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Lexicographic Extensions preserve Euclideaness

Winfried Hochstättler, Michael Wilhelmi

TL;DR

This work investigates whether Euclideanness of oriented matroid programs is preserved under extensions, focusing on lexicographic extensions. By developing tools such as the Projection Lemma, Triangle Lemma, and normalization of directed cycles, the authors show that any directed cycle arising in a lexicographic extension can be traced back to a cycle in the original matroid program, contradicting Euclideaness if the original is Euclidean. They formulate and prove two main theorems: (i) cor:lexExtStaysEucl2a, establishing Euclidean preservation for lexicographic extensions in the basic setup, and (ii) a more general Second Main Theorem showing Euclidean preservation under broader assumptions; together they conclude that a lexicographic extension of a Euclidean oriented matroid stays Euclidean. The results have implications for the behavior of Euclidean regions, boundedness, and Mandel-type properties, and lay groundwork for further study of mutations and related constructions in Euclidean oriented matroids.

Abstract

We prove that a lexicographical extension of a Euclidean oriented matroid remains Euclidean.

Lexicographic Extensions preserve Euclideaness

TL;DR

This work investigates whether Euclideanness of oriented matroid programs is preserved under extensions, focusing on lexicographic extensions. By developing tools such as the Projection Lemma, Triangle Lemma, and normalization of directed cycles, the authors show that any directed cycle arising in a lexicographic extension can be traced back to a cycle in the original matroid program, contradicting Euclideaness if the original is Euclidean. They formulate and prove two main theorems: (i) cor:lexExtStaysEucl2a, establishing Euclidean preservation for lexicographic extensions in the basic setup, and (ii) a more general Second Main Theorem showing Euclidean preservation under broader assumptions; together they conclude that a lexicographic extension of a Euclidean oriented matroid stays Euclidean. The results have implications for the behavior of Euclidean regions, boundedness, and Mandel-type properties, and lay groundwork for further study of mutations and related constructions in Euclidean oriented matroids.

Abstract

We prove that a lexicographical extension of a Euclidean oriented matroid remains Euclidean.

Paper Structure

This paper contains 19 sections, 71 theorems, 28 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Premilinaries
  3. Directing Edges in the Graph $G_f$ and Beyond
  4. Old and new cocircuits in single-element extensions
  5. Lexicographic extensions
  6. Proof of Theorem \ref{['cor:lexExtStaysEucl2a']}
  7. Sketch of proof
  8. Directions of edges of old cocircuits in single-element extensions
  9. The directions of the cocircuit graph of the lexicographic extension $(\mathcal{O} \cup p,p,f)$
  10. First Main Theorems
  11. Proof of Theorem \ref{['theorem:SecondMainTheorem']}
  12. Sketch of Proof
  13. The Projection Lemma and the Triangle Lemma
  14. Directions of Edges with New Cocircuits in Single-element Extensions
  15. Paths and Cycles in Single Element Extensions
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\mathcal{O}$ be a Euclidean oriented matroid and let $\mathcal{O} \cup p$ be a lexicographic extension of $\mathcal{O}$. Then $(\mathcal{O} \cup p,p,f)$ is a Euclidean matroid program.

Figures (6)

  • Figure 1: Projection Lemma
  • Figure 2: Constellation A
  • Figure 3: Constellation AB
  • Figure 4: Constellation B
  • Figure 5: Index Change Constellation Without Side Change
  • ...and 1 more figures

Theorems & Definitions (140)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3: Uniqueness Of Cocircuit Elimination
  • Definition 2.1
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.2: see 2, 10.1.16
  • ...and 130 more