Adjoints of matroids

Abstract

We first show that an adjoint of a loopless matroid is connected if and only if the original matroid is connected. By proving that the opposite lattice of a modular matroid is isomorphic to its extension lattice, we obtain that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice. This makes projective geometries become a key ingredient in characterizing the adjoint sequence $a{d}^{0}M,adM,a{d}^{2}M,\dots$ of a connected matroid M. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to infinite projective geometries.