 Way-below relation

In a partially ordered set, $x\ll y$ means that every nonempty directed set $D$ whose least upper bound exists and satisfies $y\le \sup D$ contains some $d$ with $x\le d$. Taking $D=\{y\}$ shows that $x\ll y$ implies $x\le y$; the converse need not hold. Some formulations quantify over the empty directed set as well; this changes the behaviour at the least element and must be specified.