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.

We show that if a first-order structure ${\cal M}$, with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then ${\cal M}$ must be interdefinable with (ℤ ,+,0) or (ℤ ,+,<,0).

The aim of this note is to determine whether certain non-o-minimal expansions of o-minimal theories which are known to be NIP, are also distal. We observe that while tame pairs of o-minimal structures and the real field with a discrete multiplicative subgroup have distal theories, dense pairs of o-minimal structures and related examples do not.

We explain which Weierstrass ${\wp}$-functions are locally definable from other ${\wp}$-functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way.

We show that the theory of the real field with a generic real power function is decidable, relative to an oracle for the rational cut of the exponent of the power function. We also show the existence of generic computable real numbers, hence providing an example of a decidable o-minimal proper expansion of the real field by an analytic function.