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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Invariantly universal analytic quasi-orders


Authors: Riccardo Camerlo, Alberto Marcone and Luca Motto Ros
Journal: Trans. Amer. Math. Soc. 365 (2013), 1901-1931
MSC (2010): Primary 03E15
Published electronically: October 11, 2012
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Abstract: We introduce the notion of an invariantly universal pair $ (S,E)$ where $ S$ is an analytic quasi-order and $ E\subseteq S$ is an analytic equivalence relation. This means that for any analytic quasi-order $ R$ there is a Borel set $ B$ invariant under $ E$ such that $ R$ is Borel bireducible with the restriction of $ S$ to $ B$. We prove a general result giving a sufficient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in different areas of mathematics, when they are paired with natural equivalence relations.


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Additional Information

Riccardo Camerlo
Affiliation: Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email: camerlo@calvino.polito.it

Alberto Marcone
Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, viale delle Scienze 206, 33100 Udine, Italy
Email: alberto.marcone@dimi.uniud.it

Luca Motto Ros
Affiliation: Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Straße 25, A-1090 Vienna, Austria
Address at time of publication: Albert-Ludwigs-Universität Freiburg, Mathematisches Institut – Abteilung für Mathematische Logik, Eckerstraße, 1, D-79104 Freiburg im Breisgau, Germany
Email: luca.motto.ros@math.uni-freiburg.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05618-2
PII: S 0002-9947(2012)05618-2
Keywords: Analytic equivalence relations, analytic quasi-orders, Borel reducibility, completeness, invariant universality, colored linear orders, dendrites, (ultrametric) Polish spaces, separable Banach spaces
Received by editor(s): March 25, 2010
Received by editor(s) in revised form: May 2, 2011
Published electronically: October 11, 2012
Additional Notes: The first author’s research was partially supported by FWF (Austrian Research Fund) through Project number P 19898-N18. The third author’s research was supported by FWF through Project number P 19898-N18. The second author’s research was partially supported by FWF through Project number P 19898-N18 and by PRIN of Italy.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.