External Direct Product Closure/General Result

Theorem

Let $\ds \struct {S, \circ} = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.

Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$ all be closed algebraic structures.

Then $\struct {S, \circ}$ is also a closed algebraic structure.

Proof

Follows directly from External Direct Product Closure.

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$\blacksquare$