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Countable 1-transitive trees

Chicot, Katie M. and Truss, John K. (2017). Countable 1-transitive trees. In: Droste, Manfred; Fuchs, László; Goldsmith, Brendan and Strüngmann, Lutz eds. Groups, Modules and Model Theory - Surveys and Recent Developments, in Memory of Rüdiger Göbel. Springer International Publishing AG, (In Press).

DOI (Digital Object Identifier) Link: https://doi.org/10.1007/978-3-319-51718-6_11
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Abstract

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 (Towards a classification of the countable 1-transitive trees: countable lower 1-transitive linear orders. arXiv:1504.03372) 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 analysing how the possible coloured branches can fit together, with particular attention to the possibilities for cones at ramification points.

Item Type: Book Section
ISBN: 3-319-51717-1, 978-3-319-51717-9
Project Funding Details:
Funded Project NameProject IDFunding Body
Not SetNot SetThe Open University (OU)
Keywords: tree, lower semilinear order; 1-transitive; cone; ramification point; coding tree
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Related URLs:
Item ID: 49490
Depositing User: Katie Chicot
Date Deposited: 30 May 2017 15:08
Last Modified: 30 May 2017 15:08
URI: http://oro.open.ac.uk/id/eprint/49490
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