# 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 | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {a \mathop {\circ {\restriction_T} } b} \mathop {\circ {\restriction_T} } c\) |

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 5.1$. Subsets closed to an operation - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 8$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $4$: Subgroups