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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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References

REFERENCES

[1]Belegradek, Oleg, Semi-bounded relations in ordered abelian groups. Model theory and applications (Belair, L.et al., editors), Quaderni di Matematica, vol. 11, Aracne, Rome, 2002. pp. 15–39.Google Scholar
[2]Berarducci, A. and Fornasiero, A., O-minimal cohomology: Finiteness and invariance results, preprint, 2007.Google Scholar
[3]Edmundo, M., Structure theorems for o-minimal expansions of groups, Annals of Pure and Applied Logic, vol. 102 (2000), no. 1–2, pp. 159–181.CrossRefGoogle Scholar
[4]Edmundo, M. and Eleftheriou, P., Definable group extensions in semi-bounded o-minimal structures, Mathematical Logic Quarterly, to appear.Google Scholar
[5]Edmundo, M., Jones, G., and Peatfield, N., O-minimal sheaf cohomology with supports, preprint. 2006.Google Scholar
[6]Edmundo, M., A uniform bound for torsion points in o-minimal expansions of groups, preprint, 2006.Google Scholar
[7]Edmundo, M. and Otero, M., Definably compact abelian groups, Journal of Mathematical Logic. vol. 4 (2004), no. 2, pp. 163–180.CrossRefGoogle Scholar
[8]Eleftheriou, P., Compact domination for groups definable in linear o-minimal structures, preprint. 2006.Google Scholar
[9]Eleftheriou, P. and Starchenko, S., Groups definable in ordered vector spaces over ordered division rings, this Journal, vol. 72 (2007). pp. 1108–1140.Google Scholar
[10]Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measure and the NIP, Journal of the American Mathematical Society, vol. 21 (2008), pp. 563–596.CrossRefGoogle Scholar
[11]Loveys, J. and Peterzil, Y., Linear o-minimal structures, Israel Journal of Mathematics, vol. 81 (1993), pp. 1–30.CrossRefGoogle Scholar
[12]Marker, D. and Steinhorn, C., Definable types in ordered structures, this Journal, vol. 51, pp. 185–198.Google Scholar
[13]Miller, Chris and Starchenko, Sergei, A growth dichotomy for o-minimal expansions of ordered groups. Transactions of the American Mathematical Society, vol. 350 (1998), no. 9, pp. 3505–3521.CrossRefGoogle Scholar
[14]Otero, M., Peterzil, Y., and Pillay, A., Groups and rings definable in o-minimal expansions of real closed fields, Bulletin of the London Mathematical Society, vol. 28 (1996), pp. 7–14.CrossRefGoogle Scholar
[15]Peterzil, Y., A structure theorem for semi-bounded sets in the reals, this Journal, vol. 57 (1992), no. 3. pp. 779–794.Google Scholar
[16]Peterzil, Y. and Pillay, A., Generic sets in definably compact groups, fundamenta Mathematicae, vol. 193 (2007), no. 2, pp. 153–170.CrossRefGoogle Scholar
[17]Peterzil, Y. and Starchenko, S., A trichotomy theorem for o-minimal structures, Proceedings of the London Mathematical Society, vol. 77 (1998), no. 3, pp. 481–523.CrossRefGoogle Scholar
[18]Pillay, Anand, On groups and fields definable in o-minimal structures, Journal of Pure and Applied Algebra, vol. 53 (1988), no. 3, pp. 239–255.CrossRefGoogle Scholar
[19]Pillay, Anand, Scowcroft, Philip, and Steinhorn, Charles, Between groups and rings, Rocky Mountain Journal of Mathematics, vol. 19 (1989), no. 3, pp. 871–885, Quadratic forms and real algebraic geometry (Corvallis, OR, 1986).CrossRefGoogle Scholar
[20]Poston, R., Defining mutiplication in o-minimal expansions of the additive reals, this Journal, vol. 60 (1995), no. 3, pp. 797–816.Google Scholar
[21]van den Dries, Lou, o-minimal structures, Logic: from foundations to applications (Staffordshire, 1993), Oxford Science Publications, Oxford University Press, New York, 1996, pp. 137–185.CrossRefGoogle Scholar
[22]van den Dries, Lou, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar