Restriction of Associative Operation is Associative

Theorem

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

Let $T \subseteq S$.

Let $\circ_T$ denote the restriction of $\circ$ to $T$.

Let the operation $\circ$ be associative on $\struct {S, \circ}$.

Then $\circ_T$ is associative on $\struct {T, \circ_T}$.

$\ds T$

$\subseteq$

$\ds S$

$\ds \leadsto \ \ $

$\ds \forall a, b, c \in T: \, $

$\ds a, b, c$

$\in$

$\ds S$

Definition of Subset

$\ds \leadsto \ \ $

$\ds a \mathop {\circ_T} \paren {b \mathop {\circ_T} c}$

$=$

$\ds a \circ \paren {b \circ c}$

Definition of Restriction of Operation

$\ds $

$=$

$\ds \paren {a \circ b} \circ c$

Semigroup Axiom $\text S 1$: Associativity

$\ds $

$=$

$\ds \paren {a \mathop {\circ_T} b} \mathop {\circ_T} c$

Definition of Restriction of Operation

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.1$. Subsets closed to an operation
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups