Assume $(G,+)$ is an (abelian) group. And we have some unit $u$ (not the unit from ring theory, but some measurement unit like meters, seconds, etc.).
Then with $G_u = \{k\ u \mid k \in G\}$ and $+_u: G_u^2 \rightarrow G_u, (x\ u,y\ u) \mapsto (x + y)\ u$:
$(G_u,+_u)$ should be an (abelian) group again (with $1_u = 1\ u$).
So I'm wondering if there is a common notation, or a simpler representation for this. I was thinking about something like $G \times \{u\}$ but technically $u + u = u$ doesn't make sense, so the canonic tuple addition doesn't work.
Note that this doesn't work for rings as e.g. $4 cm / 2 cm = 2$ which would violate closure.