The minors of matroids with an adjoint

TL;DR

This paper proves that the class of matroids possessing an adjoint, a related simple matroid with a specific lattice embedding, remains closed under taking minors, highlighting structural stability within this class.

Contribution

It establishes that the class of matroids with an adjoint is minor-closed, providing a key structural property for this class of matroids.

Findings

The class of matroids with an adjoint is minor-closed.

Provides a proof of minor-closure for this class.

Highlights structural stability of matroids with an adjoint.

Abstract

If $M$ is a matroid, then a simple matroid $M'$ with the same rank as $M$ is an adjoint of $M$ if there is an inclusion-reversing embedding $\phi$ of the lattice of flats of $M$ into the lattice of flats of $M'$ such that $\phi$ maps the hyperplanes of $M$ bijectively onto the points of $M'$. In this note, we provide a proof that the class of matroids with an adjoint is minor-closed.