Suppose I have a lattice $L$ and some subset $S \subseteq L$. What do you call the closure of $S$ under taking (finite) joins, i.e. the smallest subset $S \subseteq \overline{S} \subseteq L$ such that if $x, y \in \overline{S}$ then $x \vee y \in \overline{S}$ as well?
I googled a couple of things like "join closure" and it seems like that term is used occasionally, but there were only a small number of Google hits for it so I'm assuming there's a more common term.
I'd also like to know about notation.
Since your closure operator only makes use of $\vee$, I would called $\overline{S}$ the $\vee$-semilattice (or the join semilattice if you prefer to avoid symbols) generated by $S$. A possible notation might be $\langle S \rangle_\vee$, or simply $\langle S \rangle$ if there is no ambiguity.