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.

中文翻译：

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有限拟阵和余拟阵混合族的基分区

令 ${\mathcal{M} = (M_i \colon i\in K)}$ 是一个有限或无限族，由在公共基础集 $E$ 上的拟阵组成，每个拟阵可能是有限的或余有限的。我们证明以下 Cantor-Bernstein 类型的结果：如果有一个基集合，每个 $M_i$ 一个，覆盖集合 $E$，还有一个成对不相交的基集合，那么有一个集合划分 $E$ 的基数。我们还表明，对于任意拟阵族，这种 Cantor-Bernstein 型陈述的失败与集合论 ZFC 的公理是一致的。