This question is motivated by an attempt to understand what is going on in Tom's post from a certain point of view.

Let $\mathbb N^+$ be the free semigroup on one generator (so the positive natural numbers with addition). Is it true that the underlying semigroup of the pro-algebraic completion of $\mathbb N^+$ can be described as $\mathbb N^+\sqcup G$ where $G$ is the pro-algebraic completion of the additive group of integers, where adding an element of $\mathbb N^+$ to an element of $G$ is done by first identifying an element of $\mathbb N^+$ with its copy in $\mathbb Z\subseteq G$?

Assume the ground field is algebraically closed if you like.

Remark: If you replace pro-algebraic by profinite, then it is true.