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A Note on Extension Properties and Representations of Matroids

Michael Bamiloshin, Oriol Farràs, Carles Padró

TL;DR

The paper unifies extension-property approaches to classify small matroids and polymatroids by their representability across linear, folded-linear, algebraic, and entropic frameworks, using both theoretical results (e.g., generalized Euclidean, Ingleton-Main, Dress-Lovász, Ahlswede-Körner) and computer-aided explorations. It introduces a common information–extension framework for matroids and polymatroids, and applies it to identify new non-linearly representable matroids, with a deep focus on sparse paving matroids and tic-tac-toe configurations, including a detailed analysis of the $T^3$ family and its nine-element counterparts. The work provides human-readable proofs for several non-GE cases, enumerates 181 nine-element TTT matroids, and describes two maximal relaxations with precise representability properties across characteristics, thereby clarifying the boundary between linear, algebraic, and almost entropic matroids and illustrating duality-related limits. Overall, the results advance understanding of how extension properties constrain representations, offering computational tools and datasets that inform secret-sharing and information-theoretic perspectives in matroid theory.

Abstract

We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic representations. Iterations of those extension properties are checked for matroids on eight and nine elements by means of computer-aided explorations, finding in that way several new examples of non-linearly representable matroids. A special emphasis is made on sparse paving matroids on nine points containing the tic-tac-toe configuration. We present a new, more clear description of that family and we analyze extension properties on those matroids and their duals.

A Note on Extension Properties and Representations of Matroids

TL;DR

The paper unifies extension-property approaches to classify small matroids and polymatroids by their representability across linear, folded-linear, algebraic, and entropic frameworks, using both theoretical results (e.g., generalized Euclidean, Ingleton-Main, Dress-Lovász, Ahlswede-Körner) and computer-aided explorations. It introduces a common information–extension framework for matroids and polymatroids, and applies it to identify new non-linearly representable matroids, with a deep focus on sparse paving matroids and tic-tac-toe configurations, including a detailed analysis of the family and its nine-element counterparts. The work provides human-readable proofs for several non-GE cases, enumerates 181 nine-element TTT matroids, and describes two maximal relaxations with precise representability properties across characteristics, thereby clarifying the boundary between linear, algebraic, and almost entropic matroids and illustrating duality-related limits. Overall, the results advance understanding of how extension properties constrain representations, offering computational tools and datasets that inform secret-sharing and information-theoretic perspectives in matroid theory.

Abstract

We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic representations. Iterations of those extension properties are checked for matroids on eight and nine elements by means of computer-aided explorations, finding in that way several new examples of non-linearly representable matroids. A special emphasis is made on sparse paving matroids on nine points containing the tic-tac-toe configuration. We present a new, more clear description of that family and we analyze extension properties on those matroids and their duals.
Paper Structure (9 sections, 16 theorems, 10 equations, 3 figures, 3 tables)

This paper contains 9 sections, 16 theorems, 10 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our Results
  3. Representations of Matroids and Polymatroids
  4. Extension Properties of Matroids and Polymatroids
  5. Matroids on Eight Elements
  6. Matroids with the Tic-Tac-Toe Configuration
  7. Other Results of Computer-Aided Explorations
  8. Identifying TTT Matroids in the Existing Databases
  9. Other Matroids on Nine Elements

Key Result

Proposition 3.4

For every field $K$, the class of $K$-linearly representable polymatroids satisfies the common information property.

Figures (3)

  • Figure 1: Geometric representation of $P_8$
  • Figure 2: The affine plane over $\mathbf{F}_3$
  • Figure 3: All TTT matroids on 9 points. The labels displayed are the Bollen tags Bollen18-1 of these matroids

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • Definition 3.5
  • Proposition 3.6
  • Definition 3.7
  • Proposition 3.8
  • ...and 24 more