Subset Product within Semigroup is Associative

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Theorem

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


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


Corollary

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


Then:

\(\ds x \paren {y S}\) \(=\) \(\ds \paren {x y} S\)
\(\ds x \paren {S y}\) \(=\) \(\ds \paren {x S} y\)
\(\ds \paren {S x} y\) \(=\) \(\ds S \paren {x y}\)


Proof

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


Then:

\(\ds X \circ_\PP \paren {Y \circ_\PP Z}\) \(=\) \(\ds \set {x \circ \paren {y \circ z}: x \in X, y \in Y, z \in Z}\) Definition of Subset Product
\(\ds \) \(=\) \(\ds \set {\paren {x \circ y} \circ z: x \in X, y \in Y, z \in Z}\) Semigroup Axiom $\text S 1$: Associativity
\(\ds \) \(=\) \(\ds \paren {X \circ_\PP Y} \circ_\PP Z\) Definition of Subset Product


demonstrating that $\circ_\PP$ is associative on $\powerset S$.

$\blacksquare$


Also see


Sources