External Direct Product Associativity/Sufficient Condition
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Theorem
Let $\struct {S \times T, \circ}$ be the external direct product of the two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\circ_1$ and $\circ_2$ be associative.
Then $\circ$ is also associative.
General Result
Let $\ds \struct {S, \circ} = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.
If $\circ_1, \ldots, \circ_n$ are all associative, then so is $\circ$.
Proof
Let $\circ_1$ and $\circ_2$ be associative.
\(\ds \paren {\tuple {s_1, t_1} \circ \tuple {s_2, t_2} } \circ \tuple {s_3, t_3}\) | \(=\) | \(\ds \tuple {\paren {s_1 \circ_1 s_2} \circ_1 s_3, \paren {t_1 \circ_2 t_2} \circ_2 t_3}\) | Definition of Operation Induced by Direct Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {s_1 \circ_1 s_2 \circ_1 s_3, t_1 \circ_2 t_2 \circ_2 t_3}\) | Definition of Associative Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {s_1 \circ_1 \paren {s_2 \circ_1 s_3}, t_1 \circ_2 \paren {t_2 \circ_2 t_3} }\) | Definition of Associative Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {s_1, t_1} \circ \paren {\tuple {s_2, t_2} \circ \tuple {s_3, t_3} }\) | Definition of Operation Induced by Direct Product |
Thus $\circ$ is associative.
$\blacksquare$
Also see
- External Direct Product Commutativity
- External Direct Product Identity
- External Direct Product Inverses
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Theorem $13.1: \ 1^\circ$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.1$