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O-MINIMALISM

Published online by Cambridge University Press:  25 June 2014

HANS SCHOUTENS
HANS SCHOUTENS*
Affiliation:
DEPARTMENT OF MATHEMATICS, THE GRADUATE CENTER, CUNY, 365 FIFTH AVENUE, NEW YORK, NY 10016E-mail:hschoutens@citytech.cuny.edu
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Abstract

This paper is devoted to o-minimalism, the study of the first-order properties of o-minimal structures. The main protagonists are the pseudo-o-minimal structures, that is to say, the models of the theory of all o-minimal L-structures, but we start with a more in-depth analysis of the well-known fragment DCTC (Definable Completeness/Type Completeness), and show how it already admits many of the properties of o-minimal structures: dimension theory, monotonicity, Hardy structures, and quasi-cell decomposition, provided one replaces finiteness by discreteness in all of these. Failure of cell decomposition leads to the related notion of a eukaryote structure, and we give a criterium for a pseudo-o-minimal structure to be eukaryote.

To any pseudo-o-minimal structure, we can associate its Grothendieck ring, which in the non-o-minimal case is a nontrivial invariant. To study this invariant, we identify a third o-minimalistic property, the Discrete Pigeonhole Principle, which in turn allows us to define discretely valued Euler characteristics. As an application, we study certain analytic subsets, called Taylor sets.

Keywords

o-minimalitytame structurespseudo-o-minimalo-minimalism
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 2 , June 2014 , pp. 355 - 409
Copyright
Copyright © The Association for Symbolic Logic 2014 

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References

REFERENCES

Belegradek, Oleg, Peterzil, Ya’acov, and Wagner, Frank. Quasi-o-minimal structures, this Journal, vol. 65 (2000), no. 3, pp. 1115–1132.Google Scholar
Cluckers, Raf and Edmundo, Mário. Integration of positive constructible functions against Euler characteristic and dimension. Journal of Pure and Applied Algebra, vol. 208 (2007), no. 2, pp. 691–698.CrossRefGoogle Scholar
Denef, Jan and Dries, Louvanden. p-adic and real subanalytic sets. Annals of Mathematics, vol. 128 (1988), pp. 79–138.Google Scholar
Dickmann, Max. Elimination of quantifiers for ordered valuation rings, this Journal, vol. 52 (1987), no. 1, pp. 116–128.Google Scholar
Dolich, Alf, Miller, Chris, and Steinhorn, Charlie. Structures having o-minimal open core. Transactions of the American Mathematical Society, vol. 362 (2010), no. 3, pp. 1371–1411.CrossRefGoogle Scholar
Fornasiero, Antongiulio. Locally o-minimal structures and structures with locally o-minimal open core. Annals of Pure and Applied Logic, vol. 164 (2013), no. 3, pp. 211–229.Google Scholar
Friedman, Harvey and Miller, Chris. Expansions of o-minimal structures by fast sequences, this Journal, vol. 70 (2005), no. 2, pp. 410–418.Google Scholar
Krajíček, Jan. Uniform families of polynomial equations over a finite field and structures admitting an Euler characteristic of definable sets. Proceedings of the London Mathematical Society, vol. 81 (2000), no. 2, pp. 257–284.CrossRefGoogle Scholar
Krajíček, Jan and Scanlon, Thomas. Combinatorics with definable sets: Euler characteristics and Grothendieck rings. Bulletin of Symbolic Logic, vol. 6 (2000), no. 3, pp. 311–330.CrossRefGoogle Scholar
Macpherson, Dugald, Marker, David, and Steinhorn, Charles. Weakly o-minimal structures and real closed fields. Transactions of the American Mathematical Society, vol. 352 (2000), no. 12, pp. 5435–5483 (electronic).Google Scholar
Miller, Chris. Expansions of dense linear orders with the intermediate value property, this Journal, vol. 66 (2001), no. 4, pp. 1783–1790.Google Scholar
Miller, Chris. Tameness in expansions of the real field, Logic Colloquium ’01, Lecture Notes in Logic, vol. 20, Association for Symbolic Logic, Urbana, IL, 2005, pp. 281–316.Google Scholar
Miller, Chris. Basics of o-minimality and Hardy fields, Lecture notes on O-minimal structures and real analytic geometry, Fields Institute Communications, vol. 62, Springer, New York, 2012, pp. 43–69.Google Scholar
Miller, Chris and Speissegger, Patrick. Expansions of the real line by open sets: O-minimality and open cores. Fundamenta Mathematicae, vol. 162 (1999), no. 3, pp. 193–208.Google Scholar
Pillay, Anand. An introduction to stability theory, Oxford Logic Guides, vol. 8, The Clarendon Press, Oxford University Press, New York, 1983.Google Scholar
Rennet, Alex. The non-axiomatizability of o-minimality, ArXiv 1203.2715, 2013.Google Scholar
Schoutens, Hans. The use of ultraproducts in commutative algebra, Lecture Notes in Mathematics, vol. 1999, Springer-Verlag, New York, 2010.Google Scholar
Toffalori, Carlo and Vozoris, Kathryn. Notes on local o-minimality. Mathematical Logic Quarterly, vol. 55 (2009), no. 6, pp. 617–632.Google Scholar
Dries, Louvan den. Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, Cambridge University Press, Berlin, 1998.Google Scholar
Vozoris, Kathryn. The complex field with a predicate for the integers, Ph.D. thesis, University of Illinois at Chicago, 2007.Google Scholar