External Direct Product Identity/General Result

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Theorem

Let $\ds \struct {\SS, \circ} = \prod_{k \mathop = 1}^n S_k$ 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 $e_1, e_2, \ldots, e_n$ be the identity elements of $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$ respectively.

Then $\tuple {e_1, e_2, \ldots, e_n}$ is the identity element of $\struct {\SS, \circ}$.


Proof

Let $s := \tuple {s_1, s_2, \ldots, s_n}$ be an arbitrary element of $\struct {S_1, \circ_1} \times \struct {S_2, \circ_2} \times \cdots \times \struct {S_n, \circ_n}$.

Let $e := \tuple {e_1, e_2, \ldots, e_n}$.

Then:

\(\ds s \circ e\) \(=\) \(\ds \tuple {s_1, s_2, \ldots, s_n} \circ \tuple {e_1, e_2, \ldots, e_n}\)
\(\ds \) \(=\) \(\ds \tuple {s_1 \circ_1 e_1, s_2 \circ_2 e_2, \ldots, s_n \circ_n e_n}\) Definition of External Direct Product
\(\ds \) \(=\) \(\ds \tuple {s_1, s_2, \ldots, s_n}\) Definition of Identity Element
\(\ds \) \(=\) \(\ds s\) Definition of $s$

and:

\(\ds e \circ s\) \(=\) \(\ds \tuple {e_1, e_2, \ldots, e_n} \circ \tuple {s_1, s_2, \ldots, s_n}\)
\(\ds \) \(=\) \(\ds \tuple {e_1 \circ_1 s_1, e_2 \circ_2 s_2, \ldots, e_n \circ_n s_n}\) Definition of External Direct Product
\(\ds \) \(=\) \(\ds \tuple {s_1, s_2, \ldots, s_n}\) Definition of Identity Element
\(\ds \) \(=\) \(\ds s\) Definition of $s$

$\blacksquare$


Also see


Sources

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