Restriction of Associative Operation is Associative

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 $\ds$ $=$ $\ds \paren {a \mathop {\circ {\restriction_T} } b} \mathop {\circ {\restriction_T} } c$

$\blacksquare$