Representations of Infinite Tree Sets

Gollin, J. Pascal; Kneip, Jakob

doi:10.1007/s11083-020-09529-0

Open Access
Published: 28 April 2020

Representations of Infinite Tree Sets

J. Pascal Gollin¹ &
Jakob Kneip²

Order volume 38, pages 79–96 (2021)Cite this article

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Abstract

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order type ω + 1. Then we introduce and study a topological generalisation of infinite trees which can have limit edges, and show that every infinite tree set can be represented by the tree set admitted by a suitable such tree-like space.

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Acknowledgements

We would like to thank Nathan Bowler for greatly simplifying our proof of Theorem 4.6 by pointing out the redundancy of the extra condition in the definition of pseudo-arc (cf. Lemma 4.3).

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Authors and Affiliations

Discrete Mathematics Group, Institute for Basic Science (IBS), 55 Expo-ro, Yuseong-gu, Daejeon, 34126, Korea
J. Pascal Gollin
Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany
Jakob Kneip

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J. Pascal Gollin
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Jakob Kneip
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Correspondence to J. Pascal Gollin.

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The first author was supported by the Institute for Basic Science (IBS-R029-C1).

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Gollin, J.P., Kneip, J. Representations of Infinite Tree Sets. Order 38, 79–96 (2021). https://doi.org/10.1007/s11083-020-09529-0

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Received: 28 August 2019
Accepted: 02 April 2020
Published: 28 April 2020
Issue Date: April 2021
DOI: https://doi.org/10.1007/s11083-020-09529-0

Keywords

Trees
Tree sets
Tree-like spaces
Infinite graphs