Countable 1-Transitive Trees
Countable 1-Transitive Trees
复制标题
可数1-传递树
DOI：
10.1007/978-3-319-51718-6_11
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
J. Truss
中科院分区：
文献类型：
--
作者：
Katie Chicot;J. Truss
我们给出了三个工作的综述，关于2-传递树（Droste，Memoirs Am Math Soc 57（334）1985），关于弱2-传递树（Droste等人，Proc Lond Math Soc 58：454-494，1989），以及关于低1-传递线性序（Barbina和Chicot，Towards a classification of the countable 1-transitive trees：countable lower 1-transitive linear orders. arXiv：1504.03372），都在可数的情况下。我们从这些领导给一个完整的描述所有可数1-传递树。事实上，巴比娜和奇科特的工作是作为寻找这样一种描述的初步工作而进行的。这是因为任何1-传递树中的极大链都很容易被看作是下1-传递的，但不一定是1-传递的。事实上，必须考虑一个更复杂的设置，即相同情况的着色版本（其中“颜色”对应于各种类型的分歧点），所以我们在这里所做的主要部分是描述一大类可数着色的低1-传递线性序，并继续使用它来完成所有可数1-传递树的描述。这最后一个阶段涉及分析可能的彩色分支如何组合在一起，特别注意在分支点的圆锥体的可能性。
We give a survey of three pieces of work, on 2-transitive trees (Droste, Memoirs Am Math Soc 57(334) 1985), on weakly 2-transitive trees (Droste et al., Proc Lond Math Soc 58:454–494, 1989), and on lower 1-transitive linear orders (Barbina and Chicot, Towards a classification of the countable 1-transitive trees: countable lower 1-transitive linear orders. arXiv:1504.03372), all in the countable case. We lead on from these to give a complete description of all the countable 1-transitive trees. In fact the work of Barbina and Chicot was carried out as a preliminary to finding such a description. This is because the maximal chains in any 1-transitive tree are easily seen to be lower 1-transitive, but are not necessarily 1-transitive. In fact a more involved set-up has to be considered, namely a coloured version of the same situation (where ‘colours’ correspond to various types of ramification point), so a major part of what we do here is to describe a large class of countable coloured lower 1-transitive linear orders and go on to use this to complete the description of all countable 1-transitive trees. This final stage involves analyzing how the possible coloured branches can fit together, with particular attention to the possibilities for cones at ramification points.