Concrete self-similar representations of some subgroups of the Baer-Specker Group

In this work we study the self-similarity of infinite products of the group of the integers. In [6], the authors ask if there is a self-similar free abelian group of uncountable rank. On the other hand, in [3] is asked if the Baer-Specker group ℤ 𝜔 is self-similar (by a result of R. Baer, this group is not free abelian). We answer both questions positively.