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:
\(\ds x \circ \paren {y \circ S}\) | \(=\) | \(\ds \paren {x \circ y} \circ S\) | ||||||||||||
\(\ds x \circ \paren {S \circ y}\) | \(=\) | \(\ds \paren {x \circ S} \circ y\) | ||||||||||||
\(\ds \paren {S \circ x} \circ y\) | \(=\) | \(\ds S \circ \paren {x \circ y}\) |
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
- 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