Definition:Element is Way Below

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Let $\left({S, \preceq}\right)$ be an ordered set.

Let $x, y \in S$.

Then $x$ is way below $y$, denoted $x \ll y$, if and only if

for every directed subset $D$ of $S$
if $D$ admits a supremum and $y \preceq \sup D$
then there exists $d \in D$ such that $x \preceq d$