On the complexity of the relations of isomorphism and bi-embeddability
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- by Luca Motto Ros PDF
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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 $\mathcal {C}$.References
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Additional Information
- Luca Motto Ros
- Affiliation: Mathematisches Institut–Abteilung für Mathematische Logik, Albert-Ludwigs- Universität Freiburg, Eckerstraße, 1, D-79104 Freiburg im Breisgau, Germany
- MR Author ID: 865960
- Email: luca.motto.ros@math.uni-freiburg.de
- 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
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 309-323
- MSC (2010): Primary 03E15
- DOI: https://doi.org/10.1090/S0002-9939-2011-10896-7
- MathSciNet review: 2833542