Power Structure of Semigroup Ordered by Subsets is Ordered Semigroup
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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 $15.8$