# 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

 $\displaystyle$  $\displaystyle T \subseteq S$ $\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 $\displaystyle$ $=$ $\displaystyle \paren {a \mathop {\circ {\restriction_T} } b} \mathop {\circ {\restriction_T} } c$

$\blacksquare$