On the continuous gradability of the cut-point orders of R -trees
On the continuous gradability of the cut-point orders of R -trees
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关于R树切点阶的连续可分级性
DOI：
10.1016/j.topol.2021.107937
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发表时间：
2022
影响因子：
0.6
通讯作者：
Adam-Day S
中科院分区：
文献类型：
--
作者：
Adam-Day S
R-树是一种度量空间树，其中每个点都可以分支。Favre和Jonsson在2004年提出了以下问题：R-树下的序类是否可以用每个分支都与一个真实的区间序同构来刻画？在第一部分，我回答这个问题的否定：有一个“分支实树顺序”，这是不是“连续分级”。在第二部分中，我证明了一个分支实树序是连续分次的当且仅当每个好分层的子树是R-分次的。这一联系与集合论是把工作在第三部分回答细化的主要问题，产生几个独立的结果。例如，当κ ∈ c时，有一个分支实树序不是连续分次的，并且满足对应于κ-可分性的性质。相反地，在马丁公理下，在κ处这样的树不存在。
An R-tree is a certain kind of metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of orders underlying R-trees be characterised by the fact that every branch is order-isomorphic to a real interval? In the first part, I answer this question in the negative: there is a ‘branchwise-real tree order’which is not ‘continuously gradable’. In the second part, I show that a branchwise-real tree order is continuously gradable if and only if every well-stratified subtree is R-gradable. This link with set theory is put to work in the third part answering refinements of the main question, yielding several independence results. For example, when κ⩾ c, there is a branchwise-real tree order which is not continuously gradable, and which satisfies a property corresponding to κ-separability. Conversely, under Martin's Axiom at κ such a tree does not exist.