Axiom of Approximation in Up-Complete Semilattice

Theorem
Let $\mathscr S = \struct {S, \wedge, \preceq}$ be an up-complete meet semilattice.

Let:
 * $\forall x \in S: x^\ll$ is directed

Then:
 * $\mathscr S$ satisfies the axiom of approximation


 * $\forall x, y \in S: x \npreceq y \implies \exists u \in S: u \ll x \land u \npreceq y$
 * $\forall x, y \in S: x \npreceq y \implies \exists u \in S: u \ll x \land u \npreceq y$

Sufficient Condition
Let $\mathscr S$ satisfy the axiom of approximation.

Let $x, y \in S$ such that
 * $x \npreceq y$

By assumption:
 * $x^\ll$ is directed.

By definition of up-complete:
 * $x^\ll$ admits a supremum.

By the axiom of approximation:
 * $x = \map \sup {x^\ll}$

By definition of supremum:
 * if $y$ is upper bound for $x^\ll$, then $x \preceq y$

By definition of upper bound:
 * $\exists u \in x^\ll: u \npreceq y$

Thus by definition of way below closure:
 * $\exists u \in S: u \ll x \land u \npreceq y$

Necessary Condition
Let:
 * $\forall x, y \in S: x \npreceq y \implies \exists u \in S: u \ll x \land u \npreceq y$

Let $x \in S$.

By assumption:
 * $x^\ll$ is directed.

By definition of up-complete:
 * $x^\ll$ admits a supremum.

By Operand is Upper Bound of Way Below Closure:
 * $x$ is upper bound for $x^\ll$

We will prove that:
 * $\forall y \in S: y$ is upper bound for $x^\ll \implies x \preceq y$

Let $y \in S$ such that:
 * $y$ is upper bound for $x^\ll$


 * $x \npreceq y$
 * $x \npreceq y$

By assumption:
 * $\exists u \in S: u \ll x \land u \npreceq y$

By definition of way below closure:
 * $u \in x^\ll$

By definition of upper bound
 * $u \preceq y$

This contradicts $u \npreceq y$

Thus by definition of supremum:
 * $x = \map \sup {x^\ll}$

Thus by definition:
 * $\mathscr S$ satisfies the axiom of approximation.