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Transactions of the American Mathematical Society

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Finding bases of uncountable free abelian groups is usually difficult


Authors: Noam Greenberg, Dan Turetsky and Linda Brown Westrick
Journal: Trans. Amer. Math. Soc. 370 (2018), 4483-4508
MSC (2010): Primary 03C57; Secondary 03D60
DOI: https://doi.org/10.1090/tran/7232
Published electronically: December 14, 2017
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Abstract: We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending on the cardinality. For example, we show, under the assumption $ V=L$, that there is a first-order definable free abelian group with no first-order definable basis.


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Additional Information

Noam Greenberg
Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand
Email: greenberg@msor.vuw.ac.nz

Dan Turetsky
Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand
Email: dan.turetsky@vuw.ac.nz

Linda Brown Westrick
Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand
Address at time of publication: Department of Mathematics, University of Connecticut, 341 Mansfield Road U1009, Storrs, Connecticut 06269-1009
Email: westrick@uconn.edu

DOI: https://doi.org/10.1090/tran/7232
Received by editor(s): January 23, 2016
Received by editor(s) in revised form: March 7, 2017
Published electronically: December 14, 2017
Additional Notes: The first author was supported by the Marsden Fund, a Rutherford Discovery Fellowship from the Royal Society of New Zealand, and by the Templeton Foundation via the Turing centenary project “Mind, Mechanism and Mathematics”.
The third author was supported by the Rutherford Discovery Fellowship as a postdoctoral fellow.
Article copyright: © Copyright 2017 American Mathematical Society

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