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Returning to semi-bounded sets

  • Ya'Acov Peterzil (a1)

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.

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[3] M. Edmundo , Structure theorems for o-minimal expansions of groups, Annals of Pure and Applied Logic, vol. 102 (2000), no. 1–2, pp. 159181.

[7] M. Edmundo and M. Otero , Definably compact abelian groups, Journal of Mathematical Logic. vol. 4 (2004), no. 2, pp. 163180.

[10] E. Hrushovski , Y. Peterzil , and A. Pillay , Groups, measure and the NIP, Journal of the American Mathematical Society, vol. 21 (2008), pp. 563596.

[11] J. Loveys and Y. Peterzil , Linear o-minimal structures, Israel Journal of Mathematics, vol. 81 (1993), pp. 130.

[13] Chris Miller and Sergei Starchenko , A growth dichotomy for o-minimal expansions of ordered groups. Transactions of the American Mathematical Society, vol. 350 (1998), no. 9, pp. 35053521.

[14] M. Otero , Y. Peterzil , and A. Pillay , Groups and rings definable in o-minimal expansions of real closed fields, Bulletin of the London Mathematical Society, vol. 28 (1996), pp. 714.

[16] Y. Peterzil and A. Pillay , Generic sets in definably compact groups, fundamenta Mathematicae, vol. 193 (2007), no. 2, pp. 153170.

[17] Y. Peterzil and S. Starchenko , A trichotomy theorem for o-minimal structures, Proceedings of the London Mathematical Society, vol. 77 (1998), no. 3, pp. 481523.

[18] Anand Pillay , On groups and fields definable in o-minimal structures, Journal of Pure and Applied Algebra, vol. 53 (1988), no. 3, pp. 239255.

[19] Anand Pillay , Philip Scowcroft , and Charles Steinhorn , Between groups and rings, Rocky Mountain Journal of Mathematics, vol. 19 (1989), no. 3, pp. 871885, Quadratic forms and real algebraic geometry (Corvallis, OR, 1986).

[22] Lou van den Dries , Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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