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Base Partition for Mixed Families of Finitary and Cofinitary Matroids
Combinatorica ( IF 1.0 ) Pub Date : 2020-11-30 , DOI: 10.1007/s00493-020-4422-4
Joshua Erde , J. Pascal Gollin , Attila Joó , Paul Knappe , Max Pitz

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:

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。
更新日期:2020-11-30
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