Finitely generated algebra

Remarks

Example: the group $(\Z, +, -, 0)$ is finitely generated. Indeed $\operatorname{Sg}(\set{1})$ contains $0$, all sums $1 + \cdots + 1$, and their negatives, so it is all of $\Z$.

Non-example: the group $(\Q, +, -, 0)$ is not finitely generated. If $X = \set{p_1/q, \dots, p_k/q}$ with a common denominator $q \ge 1$, then every element of $\operatorname{Sg}(X)$ is an integer multiple of $1/q$, so $1/(2q) \notin \operatorname{Sg}(X)$.

Every finite algebra is finitely generated, with $X = A$. In model theory the same notion is used for structures in a signature with relation symbols, where the relations play no role in generation.