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Representations of Infinite Tree Sets

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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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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  • DOI: https://doi.org/10.1007/s11083-020-09529-0

Keywords

  • Trees
  • Tree sets
  • Tree-like spaces
  • Infinite graphs