Suppose I have a partial order $P$ such that every finite subset has a join.
I'm interested in subsets $S\subseteq P$ such that $a\vee b \in S \Longleftrightarrow a\in S, b\in S$. That is, not only is $S$ closed under finite joins, but also for each element of $S$, if that element is the supremum of some finite set, then all members of that set are also in $S$. I'm also interested in the analogous case where we consider all joins and not just finite ones.
Is there a name for subsets of a poset with this property, and do they come up anywhere?
The implication '$\Rightarrow$' is equivalent to $S$ being downward closed, i.e. that $a \leq b\in S$ implies $a\in S$. So such a set is an ideal.