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On o-minimal expansions of Archimedean ordered groups

  • Michael C. Laskowski (a1) and Charles Steinhorn (a2)
Abstract
Abstract

We study o-minimal expansions of Archimedean totally ordered groups. We first prove that any such expansion must be elementarily embeddable via a unique (provided some nonzero element is 0-definable) elementary embedding into a unique o-minimal expansion of the additive ordered group of real numbers . We then show that a definable function in an o-minimal expansion of enjoys good differentiability properties and use this to prove that an Archimedean real closed field is definable in any nonsemilinear expansion of . Combining these results, we obtain several restrictions on possible o-minimal expansions of arbitrary Archimedean ordered groups and in particular of the rational ordered group.

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[BW] R. P. Boas Jr., and D. V. Widder , Functions with positive differences, Duke Mathematical Journal, vol. 7 (1940), pp. 496503.

[DMM] L. van den Dries , A. Macintyre , and D. Marker , The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics, ser. 2, vol. 140 (1994), pp. 183205.

[KPS] J. F. Knight , A. Pillay , and C. Steinhorn , Definable sets in ordered structures. II, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 593605.

[Mi] C. Miller , Exponentiation is hard to avoid, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 257259.

[PS] A. Pillay and C. Steinhorn , Definable sets in ordered structures. I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565592.

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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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