Mapping Assigning to Element Its Compact Closure Preserves Infima and Directed Suprema

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

Let $C = \left({K\left({L}\right), preceq'}\right)$ be an ordered subset of $L$

where $K\left({L}\right)$ denotes the compact subset of $L$.

Let $P = \left({\mathcal P\left({K\left({L}\right)}\right), \precsim}\right)$ be an inclusion ordered set of power set of $K\left({L}\right)$.

Then there exists $f:S \to \mathcal P\left({K\left({L}\right)}\right)$ such that $f$ preserves infima and directed suprema and is an injection and $\forall x \in S: f\left({x}\right) = x^{\mathrm{compact} }$

where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.

Proof
By definitions of compact subset, compact closure, and subset:
 * $\forall x \in S: x^{\mathrm{compact} } \subseteq K\left({L}\right)$

By definition of power set:
 * $\forall x \in S: x^{\mathrm{compact} } \in \mathcal P\left({K\left({L}\right)}\right)$

Define a mapping $f:S \to \mathcal P\left({K\left({L}\right)}\right)$ such that
 * $\forall x \in S: f\left({x}\right) = x^{\mathrm{compact} }$