A growth dichotomy for o-minimal expansions of ordered groups
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- by Chris Miller and Sergei Starchenko
- Trans. Amer. Math. Soc. 350 (1998), 3505-3521
- DOI: https://doi.org/10.1090/S0002-9947-98-02288-0
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Abstract:
Let $\mathfrak {R}$ be an o-minimal expansion of a divisible ordered abelian group $(R,<,+,0,1)$ with a distinguished positive element $1$. Then the following dichotomy holds: Either there is a $0$-definable binary operation $\cdot$ such that $(R,<,+,\cdot ,0,1)$ is an ordered real closed field; or, for every definable function $f:R\to R$ there exists a $0$-definable $\lambda \in \{0\}\cup \operatorname {Aut}(R,+)$ with $\lim _{x\to +\infty }[f(x)-\lambda (x)]\in R$. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure $\mathfrak {M}:=(M,<,\dots )$ there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) $\mathfrak {M}$-definable groups with underlying set $M$.References
- L. van den Dries, Tame Topology and O-minimal Structures, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge (to appear).
- Anand Pillay and Charles Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), no. 2, 565–592. MR 833697, DOI 10.1090/S0002-9947-1986-0833697-X
- D. Marker and C. Miller, Levelled o-minimal structures, Rev. Mat. Univ. Complut. (Madrid) 10 (1997), 241–249.
- Chris Miller, A growth dichotomy for o-minimal expansions of ordered fields, Logic: from foundations to applications (Staffordshire, 1993) Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, pp. 385–399. MR 1428013
- Margarita Otero, Ya’acov Peterzil, and Anand Pillay, On groups and rings definable in o-minimal expansions of real closed fields, Bull. London Math. Soc. 28 (1996), no. 1, 7–14. MR 1356820, DOI 10.1112/blms/28.1.7
- Y. Peterzil and S. Starchenko, A trichotomy theorem for o-minimal structures, Proc. London Math. Soc. (to appear).
- Anand Pillay and Charles Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), no. 2, 565–592. MR 833697, DOI 10.1090/S0002-9947-1986-0833697-X
- Robert J. Poston, Defining multiplication in o-minimal expansions of the additive reals, J. Symbolic Logic 60 (1995), no. 3, 797–816. MR 1348994, DOI 10.2307/2275757
Bibliographic Information
- Chris Miller
- Affiliation: Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607
- Address at time of publication: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210-1174
- MR Author ID: 330760
- Email: miller@math.ohio-state.edu
- Sergei Starchenko
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235
- Address at time of publication: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- MR Author ID: 237161
- Email: starchenko.1@nd.edu
- Received by editor(s): June 5, 1996
- Additional Notes: The first author was supported by NSF Postdoctoral Fellowship No. DMS-9407549.
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 3505-3521
- MSC (1991): Primary 03C99; Secondary 06F20, 12J15, 12L12
- DOI: https://doi.org/10.1090/S0002-9947-98-02288-0
- MathSciNet review: 1491870