# Presburger sets and p-minimal fields

@article{Cluckers2003PresburgerSA,
title={Presburger sets and p-minimal fields},
author={R. Cluckers},
journal={Journal of Symbolic Logic},
year={2003},
volume={68},
pages={153 - 162}
}
• R. Cluckers
• Published 2003
• Mathematics, Computer Science
• Journal of Symbolic Logic
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… Expand

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#### References

SHOWING 1-10 OF 22 REFERENCES
Tame Topology and O-minimal Structures
• L. Dries
• Mathematics, Computer Science
• 1998
1. Some elementary results 2. Semialgebraic sets 3. Cell decomposition 4. Definable invariants: Dimension and Euler characteristic 5. The Vapnik-Chernovenkis property in o-minimal structures 6.Expand
Arithmetic and Geometric Applications of Quantifier Elimination for Valued Fields
We survey applications of quantifier elimination to number theory and algebraic geometry, focusing on results of the last 15 years. We start with the applications of p-adic quantifier elimination toExpand
• Mathematics
• 1999
Hence the semialgebraic complexity of the fibers S x ̄2y Z p : (x, y) S ́ remains bounded as the parameter x ranges over Zm p . The proof of [2, 3.32] depends on the compactness of Z p in a wayExpand
Classification of semi-algebraic $p$-adic sets up to semi-algebraic bijection
We prove that two infinite p-adic semi-algebraic sets are isomorphic (i.e. there exists a semi-algebraic bijection between them) if and only if they have the same dimension.
p-adic semi-algebraic sets and cell decomposition.
L 1. Notation. Let p denote a fixed prime number, Zp the ring of p-adic integers and Qp the field of p-adic numbers. Let K be a fixed finite field extension of Qp. For χ € K let ord χ e Z u {+ 00}Expand
On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation
• Mathematics
• 1991
Presburger's essay on the completeness and decidability of arithmetic with integer addition but without multiplication is a milestone in the history of mathematical logic and formal metatheory. TheExpand
Presburger Arithmetic and Recognizability of Sets of Natural Numbers by Automata: New Proofs of Cobham's and Semenov's Theorems
• Mathematics, Computer Science
• Ann. Pure Appl. Log.
• 1996
Let N be the set of nonnegative integers and L ⊆ N n is definable in 〈 N , +〉 if and only if every subset of N which is definability in 𝕂 N, +, L〉 is defined. Expand
Essentially periodic ordered groups
• Mathematics, Computer Science
• Ann. Pure Appl. Log.
• 2000
An essentially periodic group G is abelian; if G is discrete, then denable functions in one variable are ultimately piecewise linear, and a group such that every model elementarily equivalent to it is coset-minimal is quasi-o- Minimal (and vice versa), and its denablefunction are piecewiselinear. Expand
On Variants of o-Minimality
• Mathematics, Computer Science
• Ann. Pure Appl. Log.
• 1996
The goal here is to begin to investigate what results hold if all the structures to be considered are appropriately “minimal” expansions of some other basic relational structure, guided by the criteria that the basic structures have mathematically interesting minimal expansions and that the class of minimal expansions has some reasonable model theory analogous to that available in the motivating contexts. Expand
Quasi-o-minimal structures
• Computer Science, Mathematics
• Journal of Symbolic Logic
• 2000
It is shown that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1, and a technique to investigate quasi-O-minimal ordered groups is developed. Expand