Let $G=\Bbb Z[\frac12]/\Bbb Z$ be the (additive) Prufer 2-group and let $\Bbb Z$ be the group of integers.
Let $X=G\times \Bbb Z$ be the following product:
Let $g_n$ be the subgroups of $G$ in increasing order.
So $g_0$ is the smallest (the trivial) subgroup of $G$ and let $g_n$ be defined recursively so $g_n=\left\{x\in G,\nu_2(x)\geq-n,\right\}$
Then $X=\bigcup_n\left(g_n \times \Bbb Z^n\right)$
Is there a name for this form of product $G\times \Bbb Z$?
Is there a name for this group $X$?
It's isomorphic to the ordinal $\omega^\omega$ adjusted by a map which sends the coefficients of every term in Cantor normal form from $\Bbb N\to\Bbb Z$, with the elements of $G$ serving to indicate whether any given coefficient is positive or negative.