A THEORY OF PAIRS FOR NON-VALUATIONAL STRUCTURES

Abstract

AbstractGiven 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$\left( {\cal M} \right)$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.