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On the complexity of the relations of isomorphism and bi-embeddability

Author: Luca Motto Ros
Journal: Proc. Amer. Math. Soc. 140 (2012), 309-323
MSC (2010): Primary 03E15
Published electronically: May 16, 2011
MathSciNet review: 2833542
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Abstract: Given an $ \mathcal{L}_{\omega_1\omega}$-elementary class $ \mathcal{C}$, that is, the collection of the countable models of some $ \mathcal{L}_{\omega_1 \omega}$-sentence, denote by $ \cong_{\mathcal{C}}$ and $ \equiv_{\mathcal{C}}$ the analytic equivalence relations of, respectively, isomorphisms and bi-embeddability on $ \mathcal{C}$. Generalizing some questions of A. Louveau and C. Rosendal, in a paper by S. Friedman and L. Motto  Ros they proposed the problem of determining which pairs of analytic equivalence relations $ (E,F)$ can be realized (up to Borel bireducibility) as pairs of the form $ (\cong_{\mathcal{C}}, \equiv_{\mathcal{C}})$, $ \mathcal{C}$ some $ \mathcal{L}_{\omega_1\omega}$-elementary class (together with a partial answer for some specific cases). Here we will provide an almost complete solution to such a problem: under very mild conditions on $ E$ and $ F$, it is always possible to find such an $ \mathcal{L}_{\omega_1\omega}$-elementary class $ \mathcal{C}$.

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Luca Motto Ros
Affiliation: Mathematisches Institut–Abteilung für Mathematische Logik, Albert-Ludwigs-Universität Freiburg, Eckerstraße, 1, D-79104 Freiburg im Breisgau, Germany

Keywords: Analytic equivalence relation, isomorphism, (bi-)embeddability, Borel reducibility.
Received by editor(s): April 15, 2010
Received by editor(s) in revised form: November 10, 2010
Published electronically: May 16, 2011
Additional Notes: The author would like to thank the FWF (Austrian Research Fund) for generously supporting this research through project number P 19898-N18.
Communicated by: Julia Knight
Article copyright: © Copyright 2011 American Mathematical Society

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