Restriction of Associative Operation is Associative

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Theorem

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

Let $T \subseteq S$.


Let $T$ be closed under $\circ$.

Then $\struct {T, \circ {\restriction_T} }$ is also a semigroup, where $\circ {\restriction_T}$ is the restriction of $\circ$ to $T$.


Proof

\(\ds T\) \(\subseteq\) \(\ds S\)
\(\ds \leadsto \ \ \) \(\ds \forall a, b, c \in T: \, \) \(\ds a, b, c\) \(\in\) \(\ds S\) Definition of Subset
\(\ds \leadsto \ \ \) \(\ds a \mathop {\circ {\restriction_T} } \paren {b \mathop {\circ {\restriction_T} } c}\) \(=\) \(\ds a \circ \paren {b \circ c}\)
\(\ds \) \(=\) \(\ds \paren {a \circ b} \circ c\) as $\circ$ is associative
\(\ds \) \(=\) \(\ds \paren {a \mathop {\circ {\restriction_T} } b} \mathop {\circ {\restriction_T} } c\)

$\blacksquare$


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