Some model theory of Th ( N , · ) $\operatorname{Th}(\mathbb {N},\cdot )$

‘Skolem arithmetic’ is the complete theory T of the multiplicative monoid( N , · ) $(\mathbb {N},\cdot )$ . We give a full characterization of the ⌀ $\varnothing$ ‐definable stably embedded sets of T , showing in particular that, up to the relation of having the same definable closure, there is only one non‐trivial one: the set of squarefree elements. We then prove that T has weak elimination of imaginaries but not elimination of finite imaginaries.