A growth dichotomy for o-minimal expansions of ordered groups |
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Authors: | Chris Miller Sergei Starchenko |
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Affiliation: | Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607 ; Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235 |
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Abstract: | Let be an o-minimal expansion of a divisible ordered abelian group with a distinguished positive element . Then the following dichotomy holds: Either there is a -definable binary operation such that is an ordered real closed field; or, for every definable function there exists a -definable with . This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) -definable groups with underlying set . |
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