有限树和超限序列上的良好偏序
Well-quasi-orders on finite trees and transfinite sequences
AI总结：
本文研究了有限树和超限序列上的良好偏序，通过树同态和标签顺序的关系，改进了Friedman和Weiermann的结果，并给出了超限序列最大序型的精确界限。
摘要：
我们研究由带有来自任意良好偏序Q的叶标签的有限树组成的良好偏序（wqo），其通过保持标签顺序的树同态进行排序。这是通常的Kruskal树排序的一种变体，不涉及极小值保持。我们计算了该类wqo的精确最大序型，作为标签Q的最大序型函数。在这个过程中，我们改进了Friedman和Weiermann最近的一些结果。此外，我们展示了一种与具有有限范围的不可分解超限序列之间的对应关系，这些序列的长度小于ω^ω，元素来自wqo Q。Nash-Williams证明了任意具有有限范围的超限序列也是良好偏序的，但目前尚无方法从证明中提取最大序型的界限。Erdős和Rado为长度小于α（α<ω^ω）的序列给出了更具体的证明。利用这种对应关系，我们获得了所有具有有限范围的超限序列的精确界限，这些序列的长度小于ω^ω。
英文摘要：
We study the well-quasi-order (wqo) consisting of the set of finite trees with leaf labels coming from an arbitrary wqo $Q$, ordered by tree homomorphisms which respect the order on the labels. This is a variant of the usual Kruskal tree ordering without infima preservation. We calculate the precise maximal order types of this class of wqos as a function of the maximal order type of the labels $Q$. In the process, we sharpen some recent results of Friedman and Weiermann. Furthermore, we show a correspondence with indecomposable transfinite sequences with finite range, over elements of the wqo $Q$, of length less than $ω^ω$. Nash-Williams proved that arbitrary transfinite sequences with finite range are also well-quasi-ordered, but there are no known methods to extract bounds on the maximal order type from the proof. More concrete proofs for sequences of length less than $α$ for some $α< ω^ω$ were given by Erdős and Rado. Using the correspondence, we obtain precise bounds for the entire collection of transfinite sequences with finite range of length less than $ω^ω$.
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