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A family of rank $4$ non-algebraic matroids with pseudomodular dual

Winfried Hochstättler

TL;DR

This work addresses whether the dual of an algebraic matroid must be algebraic by constructing an infinite family of rank $4$ paving matroids $M_k$ that are non-algebraic and generalize the Tic-Tac-Toe matroid. The primal mats $M_k$ are defined with explicit circuit hyperplanes and shown non-algebraic via a Vamos configuration derived from the Ingleton–Main lemma, while their duals $M_k^*$ are proved to be pseudomodular using a detailed lattice-theoretic analysis of flats and modular pairs. Together, these results provide a broad class of examples exhibiting non-algebraic primal matroids with pseudomodular duals, contributing to the understanding of duality in algebraic matroids and the role of pseudomodularity. The paper also discusses related work and open questions, including whether some $M_k^*$ may themselves be non-algebraic and how to extend these constructions without introducing the $M_3$-like substructures. These findings illuminate the boundary between algebraic and non-algebraic matroids and raise fundamental questions about dualizability in the algebraic setting.

Abstract

The Tic-Tac-Toe matroid is a paving matroid of rank $5$ on 9 elements which is pseudomodular and whose dual is non-algebraic. It has been proposed as a possible example of an algebraic matroid whose dual is not algebraic. We present an infinite family of matroids sharing these properties and generalizing the Tic-Tac-Toe matroid.

A family of rank $4$ non-algebraic matroids with pseudomodular dual

TL;DR

This work addresses whether the dual of an algebraic matroid must be algebraic by constructing an infinite family of rank paving matroids that are non-algebraic and generalize the Tic-Tac-Toe matroid. The primal mats are defined with explicit circuit hyperplanes and shown non-algebraic via a Vamos configuration derived from the Ingleton–Main lemma, while their duals are proved to be pseudomodular using a detailed lattice-theoretic analysis of flats and modular pairs. Together, these results provide a broad class of examples exhibiting non-algebraic primal matroids with pseudomodular duals, contributing to the understanding of duality in algebraic matroids and the role of pseudomodularity. The paper also discusses related work and open questions, including whether some may themselves be non-algebraic and how to extend these constructions without introducing the -like substructures. These findings illuminate the boundary between algebraic and non-algebraic matroids and raise fundamental questions about dualizability in the algebraic setting.

Abstract

The Tic-Tac-Toe matroid is a paving matroid of rank on 9 elements which is pseudomodular and whose dual is non-algebraic. It has been proposed as a possible example of an algebraic matroid whose dual is not algebraic. We present an infinite family of matroids sharing these properties and generalizing the Tic-Tac-Toe matroid.

Paper Structure

This paper contains 5 sections, 3 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Combinatorial Properties of Algebraic Matroids
  3. The Primal Matroids
  4. The Dual Matroids
  5. Remarks and Open Problems

Key Result

Theorem 1

Let $M=(E,\mathcal{I})$ be an algebraic matroid represented by a set $E' \subseteq K$ over a field $k$, $S'$ in series in $A'\subseteq E'$ and $K$ algebraically closed. Then there exists $\beta \in K$ such that $\forall T' \subseteq A' \setminus S'$:

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1: Dress-Lovász 1987
  • Definition 4
  • Theorem 2
  • proof
  • Remark 1
  • Definition 5
  • Theorem 3
  • ...and 1 more