Is there a term to describe a preordered set $P$ in which any finite subset $S \subset P$ has at most finitely many minimal upper bounds? The preordered sets I'm studying generally aren't join-semilattices. Apologies if this is well-known.
The principle example I'm looking at is the set of vertices in a directed graph, where we take $v \leq w$ to mean that there is a directed path from $w$ to $v$.
