Direct system of abelian groups

A direct system indexed by a directed set $A$ consists of abelian groups $G_a$ and homomorphisms ${\varphi }_{ab}:G_a\to G_b$ for $a\le b$, with ${\varphi }_{aa}=1$ and ${\varphi }_{bc}{\varphi }_{ab}={\varphi }_{ac}$.

The direct limit of a direct system of abelian groups is the quotient of ${⨁}_a{G}_a$ by the relations identifying $g\in G_a$ with ${\varphi }_{ab}(g)\in G_b$ whenever $a\le b$.

For nonzero integers $a_i$ and $P_n=a_0\cdots a_{n-1}$, the compatible embeddings of stage $n$ into $\mathbb{Q}$ send $m$ to $m/P_n$. They identify the direct limit with

The additive group ${\mathbb{Q}}_{\mathrm{sq}}$ consists of the rational numbers whose reduced denominators are square-free integers. If $p_0,p_1,\dots$ lists the primes without repetition, then