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.
@article{CML_2011__3_4_587_0,
author = {Cluckers, Raf and Halupczok, Immanuel},
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 = {http://www.numdam.org/articles/10.1142/S1793744211000473/}
}
TY - JOUR AU - Cluckers, Raf AU - Halupczok, Immanuel 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 - http://www.numdam.org/articles/10.1142/S1793744211000473/ DO - 10.1142/S1793744211000473 LA - en ID - CML_2011__3_4_587_0 ER -
%0 Journal Article %A Cluckers, Raf %A Halupczok, Immanuel %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 http://www.numdam.org/articles/10.1142/S1793744211000473/ %R 10.1142/S1793744211000473 %G en %F CML_2011__3_4_587_0
Cluckers, Raf; Halupczok, Immanuel. Quantifier elimination in ordered abelian groups. Confluentes Mathematici, Tome 3 (2011) no. 4, pp. 587-615. doi : 10.1142/S1793744211000473. http://www.numdam.org/articles/10.1142/S1793744211000473/
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