Power Structure of Semigroup Ordered by Subsets is Ordered Semigroup

Theorem

Let $\struct {S, \circ}$ be a semigroup.

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.

Let $\struct {\powerset S, \circ_\PP, \subseteq}$ be the ordered structure formed from $\struct {\powerset S, \circ_\PP}$ and the subset relation.

Then $\struct {\powerset S, \circ_\PP, \subseteq}$ is an ordered semigroup.

Proof

From Power Structure of Semigroup is Semigroup, $\struct {\powerset S, \circ_\PP}$ is a semigroup.

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

It remains to be shown that $\subseteq$ is compatible with $\circ_\PP$.

This is demonstrated directly in Subset Relation is Compatible with Subset Product.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups: Exercise $15.8$