Power Set with Intersection and Superset Relation is Ordered Semigroup

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $S$ be a set and let $\powerset S$ be its power set.

Let $\struct {\powerset S, \cap, \supseteq}$ be the ordered structure formed from the set intersection operation and superset relation.


Then $\struct {\powerset S, \cap, \supseteq}$ is an ordered semigroup.


Proof

From Power Set with Intersection is Commutative Monoid, $\struct {\powerset S, \cap}$ is a fortiori a semigroup.

From Subset Relation is Ordering, $\struct {\powerset S, \subseteq}$ is an ordered set.

We have that $\supseteq$ is the dual to $\subseteq$.

Hence $\struct {\powerset S, \supseteq}$ is an ordered set.

It remains to be shown that $\supseteq$ is compatible with $\cap$.


Let $A, B \subseteq S$ be arbitrary such that $A \supseteq B$.

Thus:

\(\ds A\) \(\supseteq\) \(\ds B\)
\(\ds \leadsto \ \ \) \(\ds \forall T \subseteq S: \, \) \(\ds A \cap T\) \(\supseteq\) \(\ds B \cap T\) Set Intersection Preserves Subsets: Corollary
\(\ds \leadsto \ \ \) \(\ds \forall T \subseteq S: \, \) \(\ds T \cap A\) \(\supseteq\) \(\ds T \cap B\) Intersection is Commutative

Hence the result.

$\blacksquare$


Sources