A growth dichotomy

for o-minimal expansions of ordered groups

Authors:
Chris Miller and Sergei Starchenko

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an o-minimal expansion of a divisible ordered abelian group with a distinguished positive element . Then the following dichotomy holds: Either there is a -definable binary operation such that is an ordered real closed field; or, for every definable function there exists a -definable with . This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) -definable groups with underlying set .

**[1]**L. van den Dries,*Tame Topology and O-minimal Structures*, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge (to appear).**[2]**J. Knight, A. Pillay and C. Steinhorn,*Definable sets in ordered structures. II*, Trans. Amer. Math. Soc.**295**(1986), 593-605. MR**88b:03050b****[3]**D. Marker and C. Miller,*Levelled o-minimal structures*, Rev. Mat. Univ. Complut. (Madrid)**10**(1997), 241-249.**[4]**C. Miller,*A growth dichotomy for o-minimal expansions of ordered fields*, Logic: from Foundations to Applications, Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, pp. 385-399. MR**98a:03052****[5]**M. Otero, Y. Peterzil and A. Pillay,*On groups and rings definable in o-minimal expansions of real closed fields*, Bull. London Math. Soc.**28**(1996), 7-14. MR**96i:12006****[6]**Y. Peterzil and S. Starchenko,*A trichotomy theorem for o-minimal structures*, Proc. London Math. Soc. (to appear).**[7]**A. Pillay and C. Steinhorn,*Definable sets in ordered structures. I*, Trans. Amer. Math. Soc.**295**(1986), 565-592. MR**88b:03050a****[8]**R. Poston,*Defining multiplication in o-minimal expansions of the additive reals*, J. Symbolic Logic**60**(1995), 797-816. MR**96g:03061**

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Additional 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

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

Email:
starchenko.1@nd.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-02288-0

Received by editor(s):
June 5, 1996

Additional Notes:
The first author was supported by NSF Postdoctoral Fellowship No. DMS-9407549.

Article copyright:
© Copyright 1998
American Mathematical Society