Power Structure Operation on Set of Singleton Subsets preserves Commutativity
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Theorem
Let $\struct {S, \circ}$ be a magma.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Let $S'$ denote the set of singleton elements of $\powerset S$.
Then $\circ_\PP$ is commutative if and only if $\circ$ is commutative.
Proof
From Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets:
- $\struct {S, \circ}$ is isomorphic to $\struct {S', \circ_\PP}$
The result follows from Isomorphism Preserves Commutativity.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets: Exercise $9.5$