Power Set is Complete Lattice

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Let $S$ be a set.

Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on the power set $\powerset S$ of $S$ by the relation $\subseteq$.


$\struct {\powerset S, \subseteq}$ is a complete lattice

where for every subset $\mathbb S$ of $\powerset S$:

the infimum of $\mathbb S$ necessarily admitted by $\mathbb S$ is $\bigcap \mathbb S$.

Proof 1

From Subset Relation on Power Set is Partial Ordering, we have that $\subseteq$ is a partial ordering.

We note in passing that for any set $S$:

From Supremum of Power Set, $\powerset S$ has a supremum, that is, $S$ itself
From Infimum of Power Set, $\powerset S$ has an infimum, that is, $\O$.

These are also the maximal and minimal elements of $\powerset S$.

Let $\mathbb S$ be a subset of $\powerset S$.

Then from Union is Smallest Superset:

$\paren {\forall X \in \mathbb S: X \subseteq T} \iff \bigcup \mathbb S \subseteq T$

and from Intersection is Largest Subset:

$\paren {\forall X \in \mathbb S: T \subseteq X} \iff T \subseteq \bigcap \mathbb S$

So $\bigcap \mathbb S$ is the infimum and $\bigcup \mathbb S$ is the supremum of $\struct {\mathbb S, \subseteq}$.

Hence by definition $\powerset S$ is a complete lattice.


Proof 2

From Set is Subset of Itself:

$S \in \powerset S$

Let $\mathbb S$ be a non-empty subset of $\powerset S$.

From Intersection is Subset:

$\bigcap \mathbb S \in \powerset S$

Hence, from Set of Subsets which contains Set and Intersection of Subsets is Complete Lattice:

$\struct {\powerset S, \subseteq}$ is a complete lattice

where $\bigcap \mathbb S$ is the infimum of $\mathbb S$.


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