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Defining additive subgroups of the reals from convex subsets

Author(s): Michael A. Tychonievich
Journal: Proc. Amer. Math. Soc. 137 (2009), 3473-3476.
MSC (2000): Primary 03C64; Secondary 14P10
Posted: May 8, 2009
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Abstract: Let $ G$ be a subgroup of the additive group of real numbers and let $ C\subseteq G$ be infinite and convex in $ G$. We show that $ G$ is definable in $ (\mathbb{R},+,\cdot,C)$ and that $ {\mathbb{Z}}$ is definable if $ G$ has finite rank. This has a number of consequences for expansions of certain o-minimal structures on the real field by multiplicative groups of complex numbers.

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Additional Information:

Michael A. Tychonievich
Affiliation: Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Email: tycho@math.ohio-state.edu

DOI: 10.1090/S0002-9939-09-09914-6
PII: S 0002-9939(09)09914-6
Received by editor(s): October 1, 2008,
Received by editor(s) in revised form: December 22, 2008, and February 14, 2009
Posted: May 8, 2009
Communicated by: Julia Knight
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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