Countably generated structure

Remarks

Here countable means finite or countably infinite, so every finitely generated algebra is countably generated.

Example: the group $(\Q, +, -, 0)$ is countably generated. Take $X = \set{1/n : n \ge 1}$; then $p/q = p \cdot (1/q)$ is a sum of copies of $1/q$ or of its negative, so $\operatorname{Sg}(X) = \Q$.

Non-example: the group $(\R, +, -, 0)$ is not countably generated. For a countable $X$, each element of $\operatorname{Sg}(X)$ is the value of a term at finitely many elements of $X$. There are only countably many such values, so $\operatorname{Sg}(X)$ is countable, while $\R$ is not.

The same counting shows that, when $L$ is countable, an $L$-structure is countably generated if and only if its universe is countable.