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A THEORY OF PAIRS FOR NON-VALUATIONAL STRUCTURES
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- Journal:
- The Journal of Symbolic Logic / Volume 84 / Issue 2 / June 2019
- Published online by Cambridge University Press:
- 25 January 2019, pp. 664-683
- Print publication:
- June 2019
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- Article
- Export citation
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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.
