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