Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-23T12:16:00.866Z Has data issue: false hasContentIssue false
  • English
  • Français

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/
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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