Power Set is Complete Lattice/Proof 2

Theorem

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 $\subseteq$.


Then:

$\struct {\powerset S, \subseteq}$ 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

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 $\mathbb S$.

$\blacksquare$


Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.11 \ \text{(c) (3)}$
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