Ideals are Continuous Lattice Subframe of Power Set

Theorem
Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.

Let $I = \paren {\map {\operatorname{Ids} } L, \precsim}$ be an inclusion ordered set

where
 * $\map {\operatorname{Ids} } L$ denotes the set of all ideals in $L$,
 * $\mathord\precsim = \mathord\subseteq \cap \paren {\map {\operatorname{Ids} } L \times \map {\operatorname{Ids} } L}$

Let $P = \struct {\powerset S, \precsim'}$ be an inclusion ordered set

where
 * $\powerset S$ denotes the power set of $S$,
 * $\mathord\precsim' = \mathord\subseteq \cap \paren {\powerset S \times \powerset S}$

Then $I$ is continuous lattice subframe of $P$.

Proof
By definition of subset:
 * $\map {\operatorname{Ids} } L \subseteq \powerset S$

Then
 * $\mathord\precsim = \mathord\precsim' \cap \paren {\map {\operatorname{Ids} } L \times \map {\operatorname{Ids} } L}$

Hence $I$ is ordered subset of $P$.

Infima Inheriting
Let $A$ be a subset of $\map {\operatorname{Ids} } L$ such that:
 * $A$ admits an infimum in $P$.

By proof of Power Set is Complete Lattice:
 * $\ds \inf_P A = \bigcap A$

By Intersection of Semilattice Ideals is Ideal/Set of Sets:
 * $\ds \inf_P A \in \map {\operatorname{Ids} } L$

Thus by Infimum in Ordered Subset:
 * $A$ admits an infimum in $I$ and $\ds \inf_I A = \inf_P A$

Hence $I$ inherits infima.

Directed Suprema Inheriting
Let $D$ be a directed subset of $\map {\operatorname{Ids} } L$ such that:
 * $D$ admits a supremum in $P$.

By proof of Power Set is Complete Lattice:
 * $\ds \sup_P D = \bigcup D$

We will prove that
 * $\ds \bigcup D$ is an ideal in $L$.

Directed
Let $\ds x, y \in \bigcup D$.

By definition of union:
 * $\exists I_1 \in D: x \in I_1$

and
 * $\exists I_2 \in D: y \in I_2$

By definition of directed:
 * $\exists I \in D: I_1 \precsim I \land I_2 \precsim I$

By definition of $\precsim$:
 * $I_1 \subseteq I$ and $I_2 \subseteq I$

By definition of subset:
 * $x, y \in I$

By definition of directed:
 * $\exists z \in I: x \preceq z \land y \preceq z$

Thus by definition of union:
 * $\ds \exists z \in \bigcup D: x \preceq z \land y \preceq z$

Lower Set
Let $\ds x \in \bigcup D$, $y \in S$ such that:
 * $y \preceq x$

By definition of union:
 * $\exists I \in D: x \in I$

By definition of lower set:
 * $y \in I$

Thus by definition of union:
 * $\ds y \in \bigcup D$

Non-Empty Set
By definition of directed:
 * $D$ is non-empty and $\forall I \in D: I$ is non-empty.

Thus by definitions of non-empty set and union:
 * $\ds \bigcup D$ is non-empty.

By definition of $\operatorname{Ids}$:
 * $\ds \bigcup D \in \map {\operatorname{Ids} } L$

Hence $I$ inherits directed suprema.