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Returning to semi-bounded sets

Published online by Cambridge University Press:  12 March 2014

Ya'Acov Peterzil
Ya'Acov Peterzil*
Affiliation:
Department of Mathematics, University of Haifa, Haifa, Israel, E-mail: kobi@math.haifa.ac.il
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Abstract

An o-minimal expansion of an ordered group is called semi-bounded if there is no definable bijection between a bounded and an unbounded interval in it (equivalently, it is an expansion of the group by bounded predicates and group automorphisms). It is shown that every such structure has an elementary extension N such that either N is a reduct of an ordered vector space, or there is an o-minimal structure , with the same universe but of different language from N, with (i) Every definable set in N is definable in , and (ii) has an elementary substructure in which every bounded interval admits a definable real closed field.

As a result certain questions about definably compact groups can be reduced to either ordered vector spaces or expansions of real closed fields. Using the known results in these two settings, the number of torsion points in definably compact abelian groups in expansions of ordered groups is given. Pillay's Conjecture for such groups follows.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 2 , June 2009 , pp. 597 - 617
Copyright
Copyright © Association for Symbolic Logic 2009

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