# Subset Product within Semigroup is Associative

## Theorem

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

Then the operation $\circ_\mathcal P$ induced on the power set of $S$ is also associative.

### Corollary

Let $\struct {S, \circ}$ be a semigroup.

Then:

 $\displaystyle x \paren {y S}$ $=$ $\displaystyle \paren {x y} S$ $\displaystyle x \paren {S y}$ $=$ $\displaystyle \paren {x S} y$ $\displaystyle \paren {S x} y$ $=$ $\displaystyle S \paren {x y}$

## Proof

Let $X, Y, Z \in \powerset S$.

Then:

 $\displaystyle X \circ_\mathcal P \paren {Y \circ_\mathcal P Z}$ $=$ $\displaystyle \set {x \circ \paren {y \circ z}: x \in X, y \in Y, z \in Z}$ $\displaystyle$ $=$ $\displaystyle \set {\paren {x \circ y} \circ z: x \in X, y \in Y, z \in Z}$ $\displaystyle$ $=$ $\displaystyle \paren {X \circ_\mathcal P Y} \circ_\mathcal P Z$

demonstrating that $\circ_\mathcal P$ is associative on $\powerset S$.

$\blacksquare$