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

TL;DR

This paper introduces an infinite family of rank 4 non-algebraic matroids with pseudomodular duals, expanding understanding of matroid duality and algebraic properties beyond the specific Tic-Tac-Toe example.

Contribution

It generalizes the Tic-Tac-Toe matroid to an infinite family of matroids with similar properties, highlighting new examples of non-algebraic duals.

Findings

Infinite family of matroids with rank 4 and non-algebraic duals

Matroids share pseudomodular dual property

Generalization of the Tic-Tac-Toe matroid

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.