Way-below relation
= Way-below relation
{title2=$x\ll y$}
= Way-below
{synonym}
In a <partially ordered set>, $x\ll y$ means that every nonempty <directed set> $D$ whose <least upper bound> exists and satisfies $y\leq\sup D$ contains some $d$ with $x\leq d$. Taking $D=\{y\}$ shows that $x\ll y$ implies $x\leq y$; 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.