Subset Product within Semigroup is Associative

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

If $\circ$ is associative, then the operation $\circ_\mathcal P$ induced on the power set of $S$ is also associative.

Proof
Let $\left({S, \circ}\right)$ be a magma in which $\circ$ is associative.

Let $X, Y, Z \in \mathcal P \left({S}\right)$.

Then:


 * $X \circ_\mathcal P \left({Y \circ_\mathcal P Z}\right) = \left\{{x \circ \left({y \circ z}\right): x \in X, y \in Y, z \in Z}\right\}$


 * $\left({X \circ_\mathcal P Y}\right) \circ_\mathcal P Z = \left\{{\left({x \circ y}\right) \circ z: x \in X, y \in Y, z \in Z}\right\}$

from which follows that $\circ_\mathcal P$ is associative on $\mathcal P \left({S}\right)$.

Also see

 * Subset Product within Commutative Structure is Commutative