External Direct Product Associativity/Sufficient Condition

From ProofWiki
Jump to navigation Jump to search

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

Sources

Retrieved from "https://proofwiki.org/w/index.php?title=External_Direct_Product_Associativity/Sufficient_Condition&oldid=564931"