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Presburger sets and p-minimal fields

Published online by Cambridge University Press:  12 March 2014

Raf Cluckers
Raf Cluckers*
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium, E-mail: raf.cluckers@wis.kuleuven.ac.be, URL: http://www.wis.kuleuven.ac.be/algebra/raf/
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Abstract

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger language.

Keywords

Model theoryPresburger arithmeticp-minimal fieldselimination of imaginariesZ-groups
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 1 , March 2003 , pp. 153 - 162
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1]Belegradek, O., Peterzil, Y., and Wagner, F. O., Quasi-o-minimal structures, this Journal, vol. 65 (2000), no. 3, pp. 1115–1132.Google Scholar
[2]Cluckers, Raf, Classification of semi-algebraic p-adic sets up to semi-algebraic bijection, Journal für die reine und angewandte Mathematik, vol. 540 (2001), pp. 105–114.Google Scholar
[3]Denef, Jan, On the evaluation of certain p-adic integrals, Théorie des nombres, sémin. delange-pisot-poitou 1983–84, vol. 59, 1985, pp. 25–47.Google Scholar
[4]Denef, Jan, p-adic semi-algebraic sets and cell decomposition, Journal für die reine und angewandte Mathematik, vol. 369 (1986), pp. 154–166.Google Scholar
[5]Denef, Jan, Arithmetic and geometric applications of quantifier elimination for valued fields, MSRI Publications, vol. 39, pp. 173–198, MSRI Publications, Cambridge University Press, 2000, pp. 173–198.Google Scholar
[6]Denef, Jan and van den Dries, Lou, p-adic and real subanalytic sets, Annals of Mathematics, vol. 128 (1988), no. 1, pp. 79–138.CrossRefGoogle Scholar
[7]Haskell, Deirdre and Macpherson, Dugald, A version of o-minimality for the p-adics, this Journal, vol. 62 (1997), no. 4, pp. 1075–1092.Google Scholar
[8]Haskell, Deirdre, Macpherson, Dugald, and van den Dries, Lou, One-dimensional p-adic subanalytic sets, Journal of the London Mathematical Society, vol. 59 (1999), no. 1, pp. 1–20.Google Scholar
[9]Hodges, W., Model theory, Encyclopedia of Mathematics and Its Applications, vol. 42, Cambridge University Press, 1993.CrossRefGoogle Scholar
[10]Macpherson, Dugald and Steinhorn, Charles, On variants of o-minimality, Annals of Pure and Applied Logic, vol. 79 (1996), no. 2, pp. 165–209.CrossRefGoogle Scholar
[11]Michaux, C. and Villemaire, R., Presburger arithmetic and recognizability of sets of natural numbers by automata: New proofs of cobham's and semenov's theorems, Annals of Pure and Applied Logic, vol. 77 (1996), no. 3, pp. 251–277.CrossRefGoogle Scholar
[12]Point, F. and Wagner, F. O., Essentially periodic ordered groups, Annals of Pure and Applied Logic, vol. 105 (2000), pp. 261–291.CrossRefGoogle Scholar
[13]Presburger, M., On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation, History and Philosophy of Logic, vol. 12 (1991), no. 2, pp. 92–101.CrossRefGoogle Scholar
[14]van den Dries, Lou, Tame topology and o-minimal structures, Lecture note series, vol. 248, Cambridge University Press, 1998.CrossRefGoogle Scholar