Power Structure of Semigroup Ordered by Supersets 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 $\struct {\powerset S, \circ_\PP, \supseteq}$ be the ordered structure formed from $\struct {\powerset S, \circ_\PP}$ and the superset relation.
Then $\struct {\powerset S, \circ_\PP, \supseteq}$ is an ordered semigroup.
Proof
From Power Structure of Semigroup Ordered by Subsets is Ordered Semigroup, $\struct {\powerset S, \circ_\PP, \subseteq}$ is an ordered semigroup.
The result then follows from Dual of Ordered Semigroup is Ordered Semigroup.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups: Exercise $15.8$