论文标题
关于Aharoni-Pouzet基本交换定理的概括
On generalisations of the Aharoni-Pouzet base exchange theorem
论文作者
论文摘要
Greene-Magnanti定理指出，如果$ m $是有限的Matroid，$ b_0 $和$ b_1 $是基础，$ b_0 = \ bigCup_ {i = 1}^{n} x_i $是一个分区\ setminus x_i）\ cup y_i $是每$ i $的基础。每个$ x_i $都是单身人士的特殊情况可以作为基本过渡图中完美匹配的存在。 Pouzet猜想这在无限维矢量空间中仍然是正确的。后来，他和Aharoni肯定地回答了这一猜想，这不仅是为矢量空间，而且针对无限的曲霉。 我们证明了他们的结果两种概括。一方面，我们证明“成为单身人士”可以放松以“有限”，这在某种意义上是敏锐的，从无限套装中排除确实是必要的。另一方面，我们证明，如果$ b_0 $和$ b_1 $是基础，那么它们的有限子集之间就有一个双圈$ f $，因此$（b_0 \ setminus i）\ cup f（i）$是每个$ i $的基础。与Aharoni和Pouzet的方法相反，我们的证明完全是基本的，它们不依赖无限匹配的理论。
The Greene-Magnanti theorem states that if $ M $ is a finite matroid, $ B_0 $ and $ B_1 $ are bases and $ B_0=\bigcup_{i=1}^{n} X_i $ is a partition, then there is a partition $ B_1=\bigcup_{i=1}^{n}Y_i $ such that $ (B_0 \setminus X_i) \cup Y_i $ is a base for every $ i $. The special case where each $ X_i $ is a singleton can be rephrased as the existence of a perfect matching in the base transition graph. Pouzet conjectured that this remains true in infinite dimensional vector spaces. Later he and Aharoni answered this conjecture affirmatively not just for vector spaces but for infinite matroids. We prove two generalisations of their result. On the one hand, we show that `being a singleton' can be relaxed to `being finite' and this is sharp in the sense the exclusion of infinite sets is really necessary. On the other hand, we prove that if $ B_0$ and $ B_1 $ are bases, then there is a bijection $ F $ between their finite subsets such that $ (B_0\setminus I) \cup F(I) $ is a base for every $ I$. In contrast to the approach of Aharoni and Pouzet, our proofs are completely elementary, they do not rely on infinite matching theory.
