Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element of Directed Set First Operand Way Below Element

Theorem

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice.

Let $x, y$ be elements of $S$.

Then

$x \ll y$

if and only if

for every directed subset $D$ of $S$ such that $y \preceq \sup D$

there exists an element $d$ of $D$: $x \ll d$

Proof

Sufficient Condition

Let $x \ll y$.

Let $D$ be a directed subset of $S$ such that

$y \preceq \sup D$

By Way Below has Interpolation Property:

$\exists x' \in S: x \ll x' \land x' \ll y$

By definition of way below relation:

$\exists d \in D: x' \preceq d$

Thus by Preceding and Way Below implies Way Below and definition of reflexivity:

$x \ll d$

$\Box$

Necessary Condition

Assume that

for every directed subset $D$ of $S$ such that $y \preceq \sup D$

there exists an element $d$ of $D$: $x \ll d$

By Way Below implies Preceding:

for every directed subset $D$ of $S$ such that $y \preceq \sup D$

there exists an element $d$ of $D$: $x \preceq d$

Thus by definition of way below relation: $x \ll y$

$\blacksquare$

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_4:53