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The decomposability problem for torsion-free abelian groups is analytic-complete


Author: Kyle Riggs
Journal: Proc. Amer. Math. Soc. 143 (2015), 3631-3640
MSC (2010): Primary 03D45, 03C57
DOI: https://doi.org/10.1090/proc/12509
Published electronically: April 6, 2015
MathSciNet review: 3348804
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Abstract: We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is $ \Sigma ^0_3$-complete. However, when we consider groups of infinite rank, it becomes $ \Sigma _1^1$-complete, so it cannot be characterized by a first-order formula in the language of arithmetic.


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Additional Information

Kyle Riggs
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Address at time of publication: Department of Mathematics, Eastern Washington University, Cheney, WA 99004
Email: kriggs3@ewu.edu

DOI: https://doi.org/10.1090/proc/12509
Received by editor(s): April 2, 2013
Received by editor(s) in revised form: November 2, 2013
Published electronically: April 6, 2015
Additional Notes: The author would like to thank Steffen Lempp and Alexander Melnikov for their fruitful discussions.
Communicated by: Mirna Dzamonja
Article copyright: © Copyright 2015 American Mathematical Society

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