Structures having ominimal open core
Authors:
Alfred Dolich, Chris Miller and Charles Steinhorn
Journal:
Trans. Amer. Math. Soc. 362 (2010), 13711411
MSC (2000):
Primary 03C64; Secondary 06F20
Published electronically:
October 8, 2009
MathSciNet review:
2563733
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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 ominimal 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 ominimal open core. Specifically, the following is proved: 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 ominimal. Two examples of classes of structures that are not ominimal yet have ominimal open core are discussed: dense pairs of ominimal expansions of ordered groups, and expansions of ominimal structures by generic predicates. In particular, such structures have open core interdefinable with the original ominimal 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 ominimal open core.
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 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, 563579.
 [2]
 A. Berenstein, C. Ealy, and A. Günaydın, Thorn independence in the field of real numbers with a small multiplicative group, Ann. Pure Appl. Logic 150 (2007), no. 13, 118. MR 2370512
 [3]
 Z. Chatzidakis and A. Pillay, Generic structures and simple theories, Ann. Pure Appl. Logic 95 (1998), no. 13, 7192. MR 1650667 (2000c:03028)
 [4]
 A. Dolich, A note on weakly ominimal structures and definable completeness, Notre Dame J. Formal Logic 48 (2007), no. 2, 281292 (electronic). MR 2306397 (2008i:03042)
 [5]
 R. Dougherty and C. Miller, Definable Boolean combinations of open sets are Boolean combinations of open definable sets, Illinois J. Math. 45 (2001), no. 4, 13471350. MR 1895461 (2003c:54018)
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 L. van den Dries, The field of reals with a predicate for the powers of two, Manuscripta Math. 54 (1985), no. 12, 187195. MR 808687 (87d:03098)
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 L. van den Dries, ominimal structures, Logic: from foundations to applications (Staffordshire, 1993) Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, pp. 137185. MR 1428004 (98b:03053)
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 L. van den Dries, convexity and tame extensions. II, J. Symbolic Logic 62 (1997), no. 1, 1434. MR 1450511 (98h:03048)
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 L. van den Dries, Dense pairs of ominimal structures, Fund. Math. 157 (1998), no. 1, 6178. MR 1623615 (2000a:03058)
 [10]
 L. van den Dries, Tame topology and ominimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348 (99j:03001)
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 L. van den Dries, Limit sets in ominimal structures, Lecture Notes in Real Analytic and Algebraic Geometry, Cuvillier Verlag, Göttingen (2005), 172215, has appeared in: Proceedings of the RAAG Summer School, Lisbon 2003: Ominimal structures.
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 L. van den Dries and A. Lewenberg, convexity and tame extensions, J. Symbolic Logic 60 (1995), no. 1, 74102. MR 1324502 (96a:03048)
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 L. van den Dries and C. Miller, Geometric categories and ominimal structures, Duke Math. J. 84 (1996), no. 2, 497540. MR 1404337 (97i:32008)
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 M. Edmundo, Structure theorems for ominimal expansions of groups, Ann. Pure Appl. Logic 102 (2000), no. 12, 159181. MR 1732059 (2001k:03080)
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 S. Fratarcangeli, Elimination of imaginaries in expansions of ominimal structures by generic sets, J. Symbolic Logic 70 (2005), no. 4, 11501160. MR 2194242 (2006i:03061)
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 H. Friedman, On expansions of ominimal structures (1996), preliminary report, available at http://www.math.ohiostate.edu/foundations/manuscripts.html.
 [17]
 H. Friedman and C. Miller, Expansions of ominimal structures by fast sequences, J. Symbolic Logic 70 (2005), no. 2, 410418. MR 2140038 (2006a:03053)
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 D. Haskell and D. Macpherson, A version of ominimality for the adics, J. Symbolic Logic 62 (1997), no. 4, 10751092. MR 1618009 (99j:03028)
 [19]
 J. Knight, A. Pillay, and C. Steinhorn, Definable sets in ordered structures. II, Trans. Amer. Math. Soc. 295 (1986), no. 2, 593605. MR 833698 (88b:03050b)
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 M. Laskowski and C. Steinhorn, On ominimal expansions of Archimedean ordered groups, J. Symbolic Logic 60 (1995), no. 3, 817831. MR 1348995 (96i:03032)
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 A. Lewenberg, On elementary pairs of ominimal structures, Ph.D. thesis, University of Illinois at UrbanaChampaign, 1995.
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 A. Macintyre, Dense embeddings. I. A theorem of Robinson in a general setting, Model theory and algebra (A memorial tribute to Abraham Robinson) Lecture Notes in Math., Vol. 498, Springer, Berlin, 1975, pp. 200219. MR 0406788 (53:10574)
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 C. Miller, Expansions of dense linear orders with the intermediate value property, J. Symbolic Logic 66 (2001), no. 4, 17831790. MR 1877021 (2003j:03044)
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 C. Miller, Tameness in expansions of the real field, Logic Colloquium '01 Lect. Notes Log., vol. 20, Assoc. Symbol. Logic, Urbana, IL, 2005, pp. 281316. MR 2143901 (2006j:03049)
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 C. Miller, Definable choice in dminimal expansions of ordered groups, preliminary report, available at http://www.math.ohiostate.edu/~miller.
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 C. Miller and P. Speissegger, Expansions of the real line by open sets: ominimality and open cores, Fund. Math. 162 (1999), no. 3, 193208. MR 1736360 (2001a:03083)
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 C. Miller and S. Starchenko, A growth dichotomy for ominimal expansions of ordered groups, Trans. Amer. Math. Soc. 350 (1998), no. 9, 35053521. MR 1491870 (99e:03025)
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 C. Miller and J. Tyne, Expansions of ominimal structures by iteration sequences, Notre Dame J. Formal Logic 47 (2006), no. 1, 9399 (electronic). MR 2211185 (2006m:03065)
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 A. Onshuus, Properties and consequences of thornindependence, J. Symbolic Logic 71 (2006), no. 1, 121. MR 2210053 (2007f:03047)
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 Y. Peterzil and S. Starchenko, A trichotomy theorem for ominimal structures, Proc. London Math. Soc. (3) 77 (1998), no. 3, 481523. MR 1643405 (2000b:03123)
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 Y. Peterzil and C. Steinhorn, Definable compactness and definable subgroups of ominimal groups, J. London Math. Soc. (2) 59 (1999), no. 3, 769786. MR 1709079 (2000i:03055)
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 A. Pillay and C. Steinhorn, Definable sets in ordered structures, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 159162. MR 741730 (86c:03033)
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 A. Pillay and C. Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), no. 2, 565592. MR 833697 (88b:03050a)
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 A. Pillay and C. Steinhorn, Discrete minimal structures.
 [37]
 A. Pillay and C. Steinhorn, Definable sets in ordered structures. III, Trans. Amer. Math. Soc. 309 (1988), no. 2, 469476. MR 943306 (89i:03059)
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 S. Randriambololona, ominimal structures: Low arity versus generation, Illinois J. Math. 49 (2005), no. 2, 547558 (electronic). MR 2164352 (2006f:03066)
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 A. Robinson, Solution of a problem of Tarski, Fund. Math. 47 (1959), 179204. MR 0112841 (22:3690)
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 C. Toffalori and K. Vozoris, Notes on local ominimality (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.ohiostate.edu
Charles Steinhorn
Affiliation:
Department of Mathematics, Vassar College, Poughkeepsie, New York 12604
Email:
steinhorn@vassar.edu
DOI:
http://dx.doi.org/10.1090/S0002994709049083
Keywords:
ominimal,
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 DMS9988855. The third author was partially supported by NSF Grant DMS0070743.
Article copyright:
© Copyright 2009
American Mathematical Society
