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
- External Direct Product Associativity
- External Direct Product Commutativity
- External Direct Product Inverses
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations: Theorem $18.10: \ 2^\circ$