Quantifier elimination in ordered abelian groups
Confluentes Mathematici, Volume 3 (2011) no. 4, pp. 587-615.

We give a new proof of quantifier elimination in the theory of all ordered abelian groups in a suitable language. More precisely, this is only "quantifier elimination relative to ordered sets" in the following sense. Each definable set in the group is a union of a family of quantifier free definable sets, where the parameter of the family runs over a set definable (with quantifiers) in a sort which carries the structure of an ordered set with some additional unary predicates.

As a corollary, we find that all definable functions in ordered abelian groups are piecewise linear on finitely many definable pieces.

Published online:
DOI: 10.1142/S1793744211000473
Raf Cluckers 1; Immanuel Halupczok 1

     author = {Raf Cluckers and Immanuel Halupczok},
     title = {Quantifier elimination in ordered abelian groups},
     journal = {Confluentes Mathematici},
     pages = {587--615},
     publisher = {World Scientific Publishing Co Pte Ltd},
     volume = {3},
     number = {4},
     year = {2011},
     doi = {10.1142/S1793744211000473},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744211000473/}
AU  - Raf Cluckers
AU  - Immanuel Halupczok
TI  - Quantifier elimination in ordered abelian groups
JO  - Confluentes Mathematici
PY  - 2011
SP  - 587
EP  - 615
VL  - 3
IS  - 4
PB  - World Scientific Publishing Co Pte Ltd
UR  - https://cml.centre-mersenne.org/articles/10.1142/S1793744211000473/
DO  - 10.1142/S1793744211000473
LA  - en
ID  - CML_2011__3_4_587_0
ER  - 
%0 Journal Article
%A Raf Cluckers
%A Immanuel Halupczok
%T Quantifier elimination in ordered abelian groups
%J Confluentes Mathematici
%D 2011
%P 587-615
%V 3
%N 4
%I World Scientific Publishing Co Pte Ltd
%U https://cml.centre-mersenne.org/articles/10.1142/S1793744211000473/
%R 10.1142/S1793744211000473
%G en
%F CML_2011__3_4_587_0
Raf Cluckers; Immanuel Halupczok. Quantifier elimination in ordered abelian groups. Confluentes Mathematici, Volume 3 (2011) no. 4, pp. 587-615. doi : 10.1142/S1793744211000473. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000473/

[1] O. Belegradek, V. Verbovskiy and F. O. Wagner, Coset-minimal groups, Ann. Pure Appl. Logic 121 (2003) 113–143.

[2] R. Cluckers and I. Halupczok, Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry, 2010, to appear in Selecta Mathematica.

[3] Y. Gurevich, Elementary properties of ordered Abelian groups, Am. Math. Soc., Transl., II. Ser. 46 (1964) 165–192 (English, Russian original).

[4] , The decision problem for some algebraic theories, 1968, Doctor of Mathemat- ics dissertation.

[5] Y. Gurevich and P. H. Schmitt, The theory of ordered abelian groups does not have the independence property, Trans. Amer. Math. Soc. 284 (1984) 171–182.

[6] I. Halupczok, A language for quantifier elimination in ordered abelian groups, Séminaire de Structures Algébriques Ordonnées 2009–2010, eds. F. Delon, M. A. Dick- mann and D. Gondard, Équipe de Logique Mathématique, Vol. 85 (Université Paris 7, 2011).

[7] B. Poizat, Cours de théorie des modèles, Bruno Poizat, Lyon, 1985, Une introduc- tion à la logique mathématique contemporaine. [An introduction to contemporary mathematical logic].

[8] M. Rubin, Theories of linear order, Israel J. Math. 17 (1974) 392–443.

[9] P. H. Schmitt, Model theory of ordered abelian groups, 1982, Habilitationsschrift.

[10] , Model- and substructure-complete theories of ordered abelian groups, in Mod- els and Sets (Aachen, 1983), Lecture Notes in Math., Vol. 1103 (Springer, 1984), pp. 389–418.

[11] V. Weispfenning, Elimination of quantifiers for certain ordered and lattice-ordered abelian groups, Proc. of the Model Theory Meeting (Univ. Brussels, Brussels/Univ. Mons, Mons, 1980), Vol. 33 (1981) 131–155.

Cited by Sources: