Power Set of Semigroup under Induced Operation is Semigroup

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Theorem

Let $\left({S, \circ}\right)$ be a semigroup.

Let $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$ be the algebraic structure consisting of the power set of $S$ and the operation induced on $\mathcal P \left({S}\right)$ by $\circ$.


Then $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$ is a semigroup.


Proof

From Power Set of Magma under Induced Operation is Magma we conclude that $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$ is a magma.

It follows from Subset Product within Semigroup is Associative that $\circ_\mathcal P$ is associative in $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$.

Thus $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$ is a semigroup.

$\blacksquare$