Subset Product of Associative 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.

Corollary

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

If $\circ$ is associative, then:

• $x \left({y S}\right) = \left({x y}\right) S$
• $x \left({S y}\right) = \left({x S}\right) y$
• $\left({S x}\right) y = S \left({x y}\right)$

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 $\mathcal P \left({S}\right)$.

$\blacksquare$

Proof of Corollary

Follows directly from the definition of Subset Product with Singleton:

$x \left({y S}\right) = \left\{{x}\right\} \left({\left\{{y}\right\} S}\right)$

and so on.

$\blacksquare$

Also see

• Subset Product of Commutative is Commutative

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 6.1$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 9$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$: Exercise $\text{G}$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 2.1$: Lemma $2.3 \ \text{(i)}$