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.
$\blacksquare$