Power Structure Operation on Set of Singleton Subsets is Closed

Theorem

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

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $S$.

Let $S'$ denote the set of singleton elements of $\powerset S$.

Then the algebraic structure $\struct {S', \circ_\PP}$ is closed.

Proof

Let $A, B \in S'$.

Then:

$\exists a, b \in S: A = \set a, B = \set b$

Hence:

\(\ds A \circ_\PP B\)

\(=\)

\(\ds \set {x \circ y: x \in A, y \in B}\)

\(\ds \)

\(=\)

\(\ds \set {a \circ b}\)

\(\ds \)

\(\in\)

\(\ds S'\)

Hence the result.

$\blacksquare$

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: Exercise $9.5$