External Direct Product Commutativity/General Result

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Theorem

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 commutative, then so is $\circ$.


Proof

Suppose that, for all $k \in \N^*_n$, $\circ_k$ is commutative.

Let $\tuple {s_1, s_2, \ldots, s_n}$ and $\tuple {t_1, t_2, \ldots, t_n}$ be elements of $\struct {S_1, \circ_1} \times \struct {S_2, \circ_2} \times \cdots \times \struct {S_n, \circ_n}$.

\(\ds \tuple {s_1, s_2, \ldots, s_n} \circ \tuple {t_1, t_2, \ldots, t_n}\) \(=\) \(\ds \tuple {s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\) Definition of Operation Induced by Direct Product
\(\ds \) \(=\) \(\ds \tuple {t_1 \circ_1 s_1, t_2 \circ_2 s_2, \ldots, t_n \circ_n s_n}\) Definition of Commutative Operation
\(\ds \) \(=\) \(\ds \tuple {t_1, t_2, \ldots, t_n} \circ \tuple {s_1, s_2, \ldots, s_n}\) Definition of Operation Induced by Direct Product

Hence the result.

$\blacksquare$


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