From well to better, the space of ideals

Raphaël Carroy ¹   ; Yann Pequignot ²

¹ Dipartimento di Matematica “Giuseppe Peano” Università di Torino Via Carlo Alberto 10 10123 Torino, Italy
² Institut des systèmes d'information Quartier UNIL-Dorigny Bâtiment Internef CH-1015 Lausanne, Switzerland and Université Paris 7 Paris, France

Fundamenta Mathematicae, Tome 227 (2014) no. 3, pp. 247-270

Résumé

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.

Détail

DOI : 10.4064/fm227-3-2

Keywords: ideals quasi order wqo naturally form compact topological space which wqo embeds other nash williams barriers given uniform structure embedding cantor space prove every map barrier wqo restricts barrier uniformly continuous map therefore extends continuous map countable closed subset cantor space space ideals wqo prove shrinking further continuous map admits canonical form regard points whose image isolated par consequence obtain simple proof result better quasi orders bqo namely wqo whose set non principal ideals bqo actually bqo