Generated subalgebra

Remarks

The set $A$ itself is closed and an intersection of closed sets is closed, so $\operatorname{Sg}(X)$ is the smallest closed subset containing $X$. For finitary operations, $\operatorname{Sg}(X) = \bigcup_{k \ge 0} X_k$, where $X_0 = X$ and $X_{k+1}$ is $X_k$ together with the constants and all values $f^{\mathbf{A}}(b_1, \dots, b_n)$ with $b_1, \dots, b_n \in X_k$.

Example: in $(\Z, +, -, 0)$, $\operatorname{Sg}(\set{4, 6}) = 2\Z$. The set $2\Z$ is closed and contains $4$ and $6$; and any closed set containing $4$ and $6$ contains $6 - 4 = 2$, hence all multiples of $2$.

Non-example: the set $4\Z \cup 6\Z$ contains $\set{4, 6}$ and $0$ but is not $\operatorname{Sg}(\set{4, 6})$, because it is not closed: $4 + 6 = 10$ is not in it.

When $\sigma$ has no constants, $\operatorname{Sg}(\emptyset) = \emptyset$.