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From well to better, the space of ideals

Volume 227 / 2014

Raphaël Carroy, Yann Pequignot Fundamenta Mathematicae 227 (2014), 247-270 MSC: Primary 06A06; Secondary 06A07, 05D10, 54H05, 03E75. DOI: 10.4064/fm227-3-2

Abstract

On the one hand, the ideals of a well quasi-order (wqo) naturally form a compact topological space into which the wqo embeds. On the other hand, Nash-Williams' barriers are given a uniform structure by embedding them into the Cantor space.

We prove that every map from a barrier into a wqo restricts on a barrier to a uniformly continuous map, and therefore extends to a continuous map from a countable closed subset of the Cantor space into the space of ideals of the wqo. We then prove that, by shrinking further, any such continuous map admits a canonical form with regard to the points whose image is not isolated. \par As a consequence, we obtain a simple proof of a result on better quasi-orders (bqo); namely, a wqo whose set of non-principal ideals is a bqo is actually a bqo.

Authors

  • Raphaël CarroyDipartimento di Matematica “Giuseppe Peano”
    Università di Torino
    Via Carlo Alberto 10
    10123 Torino, Italy
    e-mail
  • Yann PequignotInstitut des systèmes d'information
    Quartier UNIL-Dorigny
    Bâtiment Internef
    CH-1015 Lausanne, Switzerland
    and
    Université Paris 7
    Paris, France
    e-mail

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