Subset Product within Semigroup is Associative

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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 $\powerset S$.

$\blacksquare$


Also see


Sources