Wild generalised truncation of infinite matroids
J. Pascal Gollin, Attila Joó
TL;DR
This work addresses the lack of a unified truncation framework for infinite matroids by introducing generalised truncations and the notion of wild truncations, extending beyond classical $n$-truncations and $(-n)$-truncations. It develops a base-axiom characterization and a suite of preorders and equivalence relations on independent sets to control truncations, then leverages Martin's Axiom to construct wild generalized truncations of suitable finitary infinite-rank matroids. The main results show that, under MA, a finitary matroid with $|E|<\mathfrak{c}$ admits $2^{\mathfrak{c}}$ wild generalised truncations, and there are $2^{\mathfrak{c}}$ pairwise non-isomorphic ones for suitable choices, highlighting substantial combinatorial richness. These findings illuminate the landscape of truncation operations in the infinite regime and raise questions about ZFC-provable phenomena for general matroids of infinite rank.
Abstract
For ${n \in \mathbb{N}}$, the $n$-truncation of a matroid $M$ of rank at least $n$ is the matroid whose bases are the $n$-element independent sets of $M$. One can extend this definition to negative integers by letting the $(-n)$-truncation be the matroid whose bases are all the sets that can be obtained by deleting $n$ elements of a base of $M$. If $M$ has infinite rank, then for distinct ${m,n \in \mathbb{Z}}$ the $m$-truncation and the $n$-truncation are distinct matroids. Inspired by the work of Bowler and Geschke on infinite uniform matroids, we provide a natural definition of generalised truncations that encompasses the notions mentioned above. We call a generalised truncation wild if it is not an $n$-truncation for any ${n \in \mathbb{Z}}$ and we prove that, under Martin's Axiom, any finitary matroid of infinite rank and size of less than continuum admits ${2^{2^{\aleph_0}}}$ wild generalised truncations.