Submission date: 9 November 2017

Abstract:

Given a weakly o-minimal structure M and its o-minimal completion \bar M, we first associate to \bar M a canonical language and then prove that Th(M) determines Th(\bar M). We then investigate the theory of the pair M^P=(\bar M ; M) 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 M^P-definable open subset of \bar M^n$ is already definable in \bar M. We give an example of a weakly o-minimal structure which interprets \bar M and show that it is not elementarily equivalent to any reduct of an o-minimal trace.

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Full text arXiv 1711.03372: pdf, ps.