In this note, we construct a distal expansion for the structure $$\left( {; + , < ,H} \right)$$, where $H \subseteq $ is a dense $Q$-vector space basis of $R$ (a so-called Hamel basis). Our construction is also an expansion of the dense pair $\left( {; + , < ,} \right)$ and has full quantifier elimination in a natural language.

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.

We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.