Definition:Element is Way Below

Definition

Let $\struct {S, \preceq}$ 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 $x \preceq d$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_3:def 1