Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Single-element extensions of matroids over skew tracts

Ting Su

TL;DR

This work extends the classical Crapo–Las Vergnas theory of single-element extensions to the setting of matroids over skew tracts by introducing Pathetic Cancellation as the precise condition under which rank-2 contractions govern extensions. It develops a robust framework of extensions and localizations for weak and, in the stringent case, strong $T$-matroids, tying the existence of extensions to rank-2 minors and modular data through modular elimination and quasi-Plücker coordinates. The paper further shows that all stringent skew hyperfields satisfy Pathetic Cancellation, yielding analogous extension characterizations for strong matroids in that setting. Collectively, these results provide a comprehensive, cryptomorphically flavored mechanism to construct and classify one-element extensions in a broad algebraic generalization of matroid theory, with explicit connections to rank-2 contractions and modular structures.

Abstract

Matroids over skew tracts provide an algebraic framework simultaneously generalizing the notions of linear subspaces, matroids, oriented matroids, phased matroids, and some other ``matroids with extra structure". A single-element extension of a matroid $\mathcal{M}$ over a skew tract $T$ is a matroid $\widetilde{\mathcal{M}}$ over $T$ obtained from $\mathcal{M}$ by adding one more element. Crapo characterized single-element extensions of ordinary matroids, and Las Vergnas characterized single-element extensions of oriented matroids, in terms of single-element extensions of their rank 2 contractions. The results of Crapo and Las Vergnas do not generalize to matroids over skew tracts, but we will show a necessary and sufficient condition on skew tracts, called Pathetic Cancellation, such that the result can generalize to weak matroids over skew tracts. Stringent skew hyperfields are a special case of skew tracts which behave in many ways like skew fields. We find a characterization of single-element extensions of strong matroids over stringent skew hyperfields.

Single-element extensions of matroids over skew tracts

TL;DR

This work extends the classical Crapo–Las Vergnas theory of single-element extensions to the setting of matroids over skew tracts by introducing Pathetic Cancellation as the precise condition under which rank-2 contractions govern extensions. It develops a robust framework of extensions and localizations for weak and, in the stringent case, strong -matroids, tying the existence of extensions to rank-2 minors and modular data through modular elimination and quasi-Plücker coordinates. The paper further shows that all stringent skew hyperfields satisfy Pathetic Cancellation, yielding analogous extension characterizations for strong matroids in that setting. Collectively, these results provide a comprehensive, cryptomorphically flavored mechanism to construct and classify one-element extensions in a broad algebraic generalization of matroid theory, with explicit connections to rank-2 contractions and modular structures.

Abstract

Matroids over skew tracts provide an algebraic framework simultaneously generalizing the notions of linear subspaces, matroids, oriented matroids, phased matroids, and some other ``matroids with extra structure". A single-element extension of a matroid over a skew tract is a matroid over obtained from by adding one more element. Crapo characterized single-element extensions of ordinary matroids, and Las Vergnas characterized single-element extensions of oriented matroids, in terms of single-element extensions of their rank 2 contractions. The results of Crapo and Las Vergnas do not generalize to matroids over skew tracts, but we will show a necessary and sufficient condition on skew tracts, called Pathetic Cancellation, such that the result can generalize to weak matroids over skew tracts. Stringent skew hyperfields are a special case of skew tracts which behave in many ways like skew fields. We find a characterization of single-element extensions of strong matroids over stringent skew hyperfields.

Paper Structure

This paper contains 21 sections, 47 theorems, 127 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Classical results for matroids and oriented matroids
  3. single-element extensions of matroid and oriented matroid
  4. Reduction of extensions to rank 2
  5. New results for matroids over skew tracts and stringent skew hyperfields
  6. Skew hyperfields and tracts
  7. Matroids over skew tracts
  8. Structure of the paper
  9. Acknowledgment
  10. Background
  11. Skew hyperfields and skew tracts
  12. Matroids over skew tracts
  13. Minors
  14. Rescaling
  15. Crapo characterization of extension
  16. ...and 6 more sections

Key Result

Theorem 1.3

(Del11) In the standard definition of matroids, the Elimination axiom can be replaced by (Modular Elimination) Let $C_1, C_2\in C^*$ such that $C_1$ and $C_2$ form a modular pair in $C^*$, and let $e \in C_1\cap C_2$. There is a member $C_3$ of $C^*$ such that $C_3\subseteq (C_1\cup C_2) -e$.

Figures (4)

  • Figure 1: The single-element extension of the oriented matroid $\mathcal{M}$ by $p$ and the single-element extension of the matroid $\underline{\mathcal{M}}$ by $p$.
  • Figure 2:
  • Figure 3: Illustration for Example \ref{['exam2']}
  • Figure 4: Collinearities for the counterexample

Theorems & Definitions (101)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Example 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 91 more