External Direct Product Commutativity/General Result

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 $\struct {S_1, \circ_1} \times \struct {S_2, \circ_2} \times \cdots \times \struct {S_n, \circ_n}$.

Hence the result.

Also see

 * External Direct Product Associativity
 * External Direct Product Identity
 * External Direct Product Inverses