The minors of matroids with an adjoint
Kevin Grace
TL;DR
This paper proves that the class of matroids possessing an adjoint is minor-closed. It leverages adjoint maps that reverse inclusion on flats and bijectively map hyperplanes to points, and develops two constructions to propagate adjoints through minors: a contraction-compatible adjoint $\,\phi_{/C}$ when contracting $C$, and a deletion-compatible adjoint $\,\phi_{\backslash D}$ when deleting a coindependent set $D$. By showing that $M/C$ has an adjoint and that the deletion step preserves adjoints via $\,\phi_{\backslash D}$, the authors obtain an adjoint for any minor $M/C\backslash D$. This establishes minor-closure of the adjoint property and provides explicit procedures for constructing adjoints of minors; they also discuss how duality does not generally preserve the adjoint property. The results enhance understanding of adjoint stability under standard matroid operations and enable systematic adjoint construction for minors.
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 $φ$ of the lattice of flats of $M$ into the lattice of flats of $M'$ such that $φ$ 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.