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THE WADGE ORDER ON THE SCOTT DOMAIN IS NOT A WELL-QUASI-ORDER

Published online by Cambridge University Press:  29 August 2019

JACQUES DUPARC  and
LOUIS VUILLEUMIER
JACQUES DUPARC
Affiliation:
DEPARTMENT OF OPERATIONS (DO) UNIVERSITY OF LAUSANNE (UNIL) QUARTIER UNIL-CHAMBERONNE, BÂTIMENT ANTHROPOLE 1015 LAUSANNE, SWITZERLAND E-mail: jacques.duparc@unil.ch
LOUIS VUILLEUMIER
Affiliation:
DEPARTMENT OF OPERATIONS (DO) UNIVERSITY OF LAUSANNE (UNIL) QUARTIER UNIL-CHAMBERONNE, BÂTIMENT ANTHROPOLE 1015 LAUSANNE, SWITZERLAND and RESEARCH INSTITUTE ON THE FOUNDATIONS OF COMPUTER SCIENCE (IRIF) PARIS DIDEROT UNIVERSITY (PARIS 7), SORBONNE PARIS CITÉ 8 PLACE AURÉLIE NEMOURS, BÂTIMENT SOPHIE GERMAIN 75205 PARIS CEDEX 13, FRANCE E-mail: louis.vuilleumier.1@unil.chE-mail: louis.vuilleumier@etu.univ-paris-diderot.fr
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Abstract

We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets $\mathbb{P}_{emb} $ equipped with the order induced by homomorphisms is embedded into the Wadge order on the $\Delta _2^0 $-degrees of the Scott domain. We then show that $\mathbb{P}_{emb} $ admits both infinite strictly decreasing chains and infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the $\Delta _2^0 $-degrees of the Scott domain.

Keywords

03B7003D3003E1554H0568Q15descriptive set theoryquasi-Polish spaceScott domainWadge reducibilitywell-quasi-order

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Type
Articles
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The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 300 - 324
Copyright
Copyright © The Association for Symbolic Logic 2019 

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