Ideals are Continuous Lattice Subframe of Power Set

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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.

$\Box$


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:

$\ds \exists z \in \bigcup D: x \preceq z \land y \preceq z$

$\Box$


Lower Section

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 section:

$y \in I$

Thus by definition of union:

$\ds y \in \bigcup D$

$\Box$


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.

$\Box$


By definition of $\operatorname{Ids}$:

$\ds \bigcup D \in \map {\operatorname{Ids} } L$

Hence $I$ inherits directed suprema.

$\blacksquare$


Sources