Restriction of Associative Operation is Associative
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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 $\struct {T, \circ_T}$.
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 | |||||||||
\(\ds \leadsto \ \ \) | \(\ds a \mathop {\circ_T} \paren {b \mathop {\circ_T} c}\) | \(=\) | \(\ds a \circ \paren {b \circ c}\) | 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 |
$\blacksquare$
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): 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