Power Structure of Magma is Magma

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Theorem

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

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


Then $\struct {\powerset S, \circ_\PP}$ is a magma.

That is, $\circ_\PP$ is closed in $\powerset S$.


Proof

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

Let $A, B \subseteq S$.

\(\ds \forall a \in A, b \in B: \, \) \(\ds a \circ b\) \(\in\) \(\ds S\) Definition of Magma
\(\ds \leadsto \ \ \) \(\ds A \circ B\) \(\subseteq\) \(\ds S\) Definition of Subset Product
\(\ds \leadsto \ \ \) \(\ds A \circ_\PP B\) \(\subseteq\) \(\ds S\) Definition of Operation Induced on $\powerset S$ by $\circ$
\(\ds \leadsto \ \ \) \(\ds A \circ_\PP B\) \(\in\) \(\ds \powerset S\) Definition of Power Set

Thus $\struct {\powerset S, \circ_\PP}$ is a magma.

$\blacksquare$