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A THEORY OF PAIRS FOR NON-VALUATIONAL STRUCTURES

Published online by Cambridge University Press:  25 January 2019

ELITZUR BAR-YEHUDA ,
ASSAF HASSON  and
YA’ACOV PETERZIL
ELITZUR BAR-YEHUDA
Affiliation:
DEPARTMENT OF MATHEMATICS BEN GURION UNIVERSITY OF THE NEGEV BE’ER SEHVA, ISRAELE-mail: elitzur.by@gmail.com
ASSAF HASSON
Affiliation:
DEPARTMENT OF MATHEMATICS BEN GURION UNIVERSITY OF THE NEGEV BE’ER SEHVA, ISRAELE-mail: hassonas@math.bgu.ac.ilURL: http://www.math.bgu.ac.il/∼hasson/
YA’ACOV PETERZIL
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF HAIFA HAIFA, ISRAELE-mail: kobi@math.haifa.ac.ilURL: http://math.haifa.ac.il/kobi/
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Abstract

Given a weakly o-minimal structure ${\cal M}$ and its o-minimal completion $\bar{{\cal M}}$, we first associate to $\bar{{\cal M}}$ a canonical language and then prove that Th$\left( {\cal M} \right)$ determines $Th\left( {\bar{{\cal M}}} \right)$. We then investigate the theory of the pair $\left( {\bar{{\cal M}},{\cal M}} \right)$ in the spirit of the theory of dense pairs of o-minimal structures, and prove, among other results, that it is near model complete, and every definable open subset of ${\bar{M}^n}$ is already definable in $\bar{{\cal M}}$.

We give an example of a weakly o-minimal structure interpreting $\bar{{\cal M}}$ and show that it is not elementarily equivalent to any reduct of an o-minimal trace.

Keywords

03C64weakly o-minimal structuresnon-valuationaldense pairso-minimal traces
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 664 - 683
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

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