I was reading this question and I saw the answer in the comments regarding a substructure generated by 0. I have tried searching for a while online but clearly I'm not very good at it because I cant seem to find a clear answer as to what this means?
Thanks!
If $\mathcal{A}$ is a structure with universe (or underlying set, or domain) $A$ and $X\subseteq A$, then the substructure of $\mathcal{A}$ generated by $X$ is$^1$ the smallest substructure of $\mathcal{A}$ whose universe contains $X$. (See e.g. here.)
We often abuse notation when talking about singletons: "the substructure of $\mathcal{A}$ generated by $a$" is just another name for the substructure of $\mathcal{A}$ generated by $\{a\}$. A similar abuse often happens with finite sets: "the substructure of $\mathcal{A}$ generated by $a_1,...,a_k$" is just another name for the substructure of $\mathcal{A}$ generated by $\{a_1,...,a_k\}$.
For example, taking $\mathcal{A}=(\mathbb{R};+)$, we have:
The substructure of $\mathcal{A}$ generated by $0$ is just $(\{0\}; +)$.
The substructure of $\mathcal{A}$ generated by $1$ is the somewhat more complicated $(\mathbb{N}_{>0};+)$. More generally, the substructure of $\mathcal{A}$ generated by $r\in\mathbb{R}$ is $$(\{kr: k\in\mathbb{N}_{>0}\}; +).$$
The substructure of $\mathcal{A}$ generated by $3$ and $5$ is the smallest substructure of $\mathcal{A}$ with universe $\supseteq\{3,5\}$; this turns out to be $(\{3,5,6\}\cup\mathbb{N}_{>7}; +)$.
$^1$Technically this is only guaranteed to exist if $X\not=\emptyset$ or if we allow empty structures.