Structures having o-minimal open core

Authors:
Alfred Dolich, Chris Miller and Charles Steinhorn

Journal:
Trans. Amer. Math. Soc. **362** (2010), 1371-1411

MSC (2000):
Primary 03C64; Secondary 06F20

Published electronically:
October 8, 2009

MathSciNet review:
2563733

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions under which an expansion of a densely ordered group has an o-minimal open core. Specifically, the following is proved:

Two examples of classes of structures that are not o-minimal yet have o-minimal open core are discussed: dense pairs of o-minimal expansions of ordered groups, and expansions of o-minimal structures by generic predicates. In particular, such structures have open core interdefinable with the original o-minimal structure. These examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane, a phenomenon that can occur in dense pairs but not in expansions by generic predicates. The property of having no dense graphs is examined and related to uniform finiteness, definable completeness, and having o-minimal open core.Let be an expansion of a densely ordered group that is definably complete and satisfies the uniform finiteness property. Then the open core of is o-minimal.

**[1]**O. Belegradek and B. Zilber,*The model theory of the field of reals with a subgroup of the unit circle*, J. Lond. Math. Soc. (2)**78**(2008), no. 3, 563-579.**[2]**Alexander Berenstein, Clifton Ealy, and Ayhan Günaydın,*Thorn independence in the field of real numbers with a small multiplicative group*, Ann. Pure Appl. Logic**150**(2007), no. 1-3, 1–18. MR**2370512**, 10.1016/j.apal.2007.06.002**[3]**Z. Chatzidakis and A. Pillay,*Generic structures and simple theories*, Ann. Pure Appl. Logic**95**(1998), no. 1-3, 71–92. MR**1650667**, 10.1016/S0168-0072(98)00021-9**[4]**Alfred Dolich,*A note on weakly o-minimal structures and definable completeness*, Notre Dame J. Formal Logic**48**(2007), no. 2, 281–292 (electronic). MR**2306397**, 10.1305/ndjfl/1179323268**[5]**Randall Dougherty and Chris Miller,*Definable Boolean combinations of open sets are Boolean combinations of open definable sets*, Illinois J. Math.**45**(2001), no. 4, 1347–1350. MR**1895461****[6]**Lou van den Dries,*The field of reals with a predicate for the powers of two*, Manuscripta Math.**54**(1985), no. 1-2, 187–195. MR**808687**, 10.1007/BF01171706**[7]**Lou van den Dries,*o-minimal structures*, Logic: from foundations to applications (Staffordshire, 1993) Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, pp. 137–185. MR**1428004****[8]**Lou van den Dries,*𝑇-convexity and tame extensions. II*, J. Symbolic Logic**62**(1997), no. 1, 14–34. MR**1450511**, 10.2307/2275729**[9]**Lou van den Dries,*Dense pairs of o-minimal structures*, Fund. Math.**157**(1998), no. 1, 61–78. MR**1623615****[10]**Lou van den Dries,*Tame topology and o-minimal structures*, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR**1633348****[11]**L. van den Dries,*Limit sets in o-minimal structures*, Lecture Notes in Real Analytic and Algebraic Geometry, Cuvillier Verlag, Göttingen (2005), 172-215, has appeared in: Proceedings of the RAAG Summer School, Lisbon 2003: O-minimal structures.**[12]**Lou van den Dries and Adam H. Lewenberg,*𝑇-convexity and tame extensions*, J. Symbolic Logic**60**(1995), no. 1, 74–102. MR**1324502**, 10.2307/2275510**[13]**Lou van den Dries and Chris Miller,*Geometric categories and o-minimal structures*, Duke Math. J.**84**(1996), no. 2, 497–540. MR**1404337**, 10.1215/S0012-7094-96-08416-1**[14]**Mario J. Edmundo,*Structure theorems for o-minimal expansions of groups*, Ann. Pure Appl. Logic**102**(2000), no. 1-2, 159–181. MR**1732059**, 10.1016/S0168-0072(99)00043-3**[15]**Sergio Fratarcangeli,*Elimination of imaginaries in expansions of o-minimal structures by generic sets*, J. Symbolic Logic**70**(2005), no. 4, 1150–1160. MR**2194242**, 10.2178/jsl/1129642120**[16]**H. Friedman,*On expansions of o-minimal structures*(1996), preliminary report, available at`http://www.math.ohio-state.edu/foundations/manuscripts.html`.**[17]**Harvey Friedman and Chris Miller,*Expansions of o-minimal structures by fast sequences*, J. Symbolic Logic**70**(2005), no. 2, 410–418. MR**2140038**, 10.2178/jsl/1120224720**[18]**Deirdre Haskell and Dugald Macpherson,*A version of o-minimality for the 𝑝-adics*, J. Symbolic Logic**62**(1997), no. 4, 1075–1092. MR**1618009**, 10.2307/2275628**[19]**Julia F. Knight, Anand Pillay, and Charles Steinhorn,*Definable sets in ordered structures. II*, Trans. Amer. Math. Soc.**295**(1986), no. 2, 593–605. MR**833698**, 10.1090/S0002-9947-1986-0833698-1**[20]**Michael C. Laskowski,*Vapnik-Chervonenkis classes of definable sets*, J. London Math. Soc. (2)**45**(1992), no. 2, 377–384. MR**1171563**, 10.1112/jlms/s2-45.2.377**[21]**Michael C. Laskowski and Charles Steinhorn,*On o-minimal expansions of Archimedean ordered groups*, J. Symbolic Logic**60**(1995), no. 3, 817–831. MR**1348995**, 10.2307/2275758**[22]**A. Lewenberg,*On elementary pairs of o-minimal structures*, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1995.**[23]**Angus Macintyre,*Dense embeddings. I. A theorem of Robinson in a general setting*, Model theory and algebra (A memorial tribute to Abraham Robinson), Springer, Berlin, 1975, pp. 200–219. Lecture Notes in Math., Vol. 498. MR**0406788****[24]**Dugald Macpherson, David Marker, and Charles Steinhorn,*Weakly o-minimal structures and real closed fields*, Trans. Amer. Math. Soc.**352**(2000), no. 12, 5435–5483 (electronic). MR**1781273**, 10.1090/S0002-9947-00-02633-7**[25]**Chris Miller,*Expansions of dense linear orders with the intermediate value property*, J. Symbolic Logic**66**(2001), no. 4, 1783–1790. MR**1877021**, 10.2307/2694974**[26]**Chris Miller,*Tameness in expansions of the real field*, Logic Colloquium ’01, Lect. Notes Log., vol. 20, Assoc. Symbol. Logic, Urbana, IL, 2005, pp. 281–316. MR**2143901****[27]**C. Miller,*Definable choice in d-minimal expansions of ordered groups*, preliminary report, available at`http://www.math.ohio-state.edu/~miller`.**[28]**Chris Miller and Patrick Speissegger,*Expansions of the real line by open sets: o-minimality and open cores*, Fund. Math.**162**(1999), no. 3, 193–208. MR**1736360****[29]**Chris Miller and Sergei Starchenko,*A growth dichotomy for o-minimal expansions of ordered groups*, Trans. Amer. Math. Soc.**350**(1998), no. 9, 3505–3521. MR**1491870**, 10.1090/S0002-9947-98-02288-0**[30]**Chris Miller and James Tyne,*Expansions of o-minimal structures by iteration sequences*, Notre Dame J. Formal Logic**47**(2006), no. 1, 93–99. MR**2211185**, 10.1305/ndjfl/1143468314**[31]**Alf Onshuus,*Properties and consequences of thorn-independence*, J. Symbolic Logic**71**(2006), no. 1, 1–21. MR**2210053**, 10.2178/jsl/1140641160**[32]**Ya’acov Peterzil and Sergei Starchenko,*A trichotomy theorem for o-minimal structures*, Proc. London Math. Soc. (3)**77**(1998), no. 3, 481–523. MR**1643405**, 10.1112/S0024611598000549**[33]**Ya’acov Peterzil and Charles Steinhorn,*Definable compactness and definable subgroups of o-minimal groups*, J. London Math. Soc. (2)**59**(1999), no. 3, 769–786. MR**1709079**, 10.1112/S0024610799007528**[34]**Anand Pillay and Charles Steinhorn,*Definable sets in ordered structures*, Bull. Amer. Math. Soc. (N.S.)**11**(1984), no. 1, 159–162. MR**741730**, 10.1090/S0273-0979-1984-15249-2**[35]**Anand Pillay and Charles Steinhorn,*Definable sets in ordered structures. I*, Trans. Amer. Math. Soc.**295**(1986), no. 2, 565–592. MR**833697**, 10.1090/S0002-9947-1986-0833697-X**[36]**A. Pillay and C. Steinhorn,*Discrete -minimal structures*.**[37]**Anand Pillay and Charles Steinhorn,*Definable sets in ordered structures. III*, Trans. Amer. Math. Soc.**309**(1988), no. 2, 469–476. MR**943306**, 10.1090/S0002-9947-1988-0943306-9**[38]**Serge Randriambololona,*o-minimal structures: low arity versus generation*, Illinois J. Math.**49**(2005), no. 2, 547–558 (electronic). MR**2164352****[39]**A. Robinson,*Solution of a problem of Tarski*, Fund. Math.**47**(1959), 179–204. MR**0112841****[40]**C. Toffalori and K. Vozoris,*Notes on local o-minimality*(2008), to appear in MLQ Math. Log. Q., available at`http://www.logique.jussieu.fr/modnet/Home/`.**[41]**B. Zilber,*Complex roots of unity on the real plane*(2003), preprint, available at`http://www.maths.ox.ac.uk/~zilber/publ.html`.

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Additional Information

**Alfred Dolich**

Affiliation:
Department of Mathematics and Computer Science, Chicago State University, Chicago, Illinois 60628

Address at time of publication:
Department of Mathematics, East Stroudsberg University, East Stroudsberg, Pennsylvania 18301

Email:
adolich@csu.edu

**Chris Miller**

Affiliation:
Department of Mathematics, 231 West 18th Avenue, The Ohio State University, Columbus, Ohio 43210

Email:
miller@math.ohio-state.edu

**Charles Steinhorn**

Affiliation:
Department of Mathematics, Vassar College, Poughkeepsie, New York 12604

Email:
steinhorn@vassar.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-09-04908-3

Keywords:
o-minimal,
open core,
ordered group,
ordered field,
uniform finiteness,
definably complete,
exchange property,
definable Skolem functions,
elimination of imaginaries,
independence property,
atomic model,
dense pair,
generic predicate,
tame pair

Received by editor(s):
January 7, 2008

Published electronically:
October 8, 2009

Additional Notes:
The second author was partially supported by NSF Grant DMS-9988855. The third author was partially supported by NSF Grant DMS-0070743.

Article copyright:
© Copyright 2009
American Mathematical Society