Base partition for mixed families of finitary and cofinitary matroids

TL;DR

This paper proves a Cantor-Bernstein-type theorem for families of finitary and cofinitary matroids, showing conditions for partitioning the ground set into bases and discussing set-theoretic consistency issues.

Contribution

It establishes a new partitioning theorem for mixed matroid families and explores its limitations within set theory.

Findings

Proves a Cantor-Bernstein-type result for matroid families covering the ground set.

Shows the theorem's failure is consistent with ZFC set theory.

Identifies conditions under which bases can partition the ground set.

Abstract

Let ${\mathcal{M} = (M_i \colon i\in K)}$ be a finite or infinite family consisting of matroids on a common ground set $E$ each of which may be finitary or cofinitary. We prove the following Cantor-Bernstein-type result: If there is a collection of bases, one for each $M_i$, which covers the set $E$, and also a collection of bases which is pairwise disjoint, then there is a collection of bases which partitions $E$. We also show that the failure of this Cantor-Bernstein-type statement for arbitrary matroid families is consistent relative to the axioms of set theory ZFC.