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

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

$\blacksquare$


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