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 thatTh$\left$determines$Th\left$. We then investigate the theory of the pair$\left$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.