Subset Product within Semigroup is Associative

Theorem

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

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

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

Then:

$\ds x \circ \paren {y \circ S}$

$=$

$\ds \paren {x \circ y} \circ S$

$\ds x \circ \paren {S \circ y}$

$=$

$\ds \paren {x \circ S} \circ y$

$\ds \paren {S \circ x} \circ y$

$=$

$\ds S \circ \paren {x \circ y}$

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 $\powerset S$.

$\blacksquare$

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{G}$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 41.1$ Multiplication of subsets of a group