External Direct Product Identity
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Theorem
Let $\struct {S \times T, \circ}$ be the external direct product of two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Then:
- $\struct {S, \circ_1}$ has identity element $e_S$ and $\struct {T, \circ_2}$ has identity element $e_T$
- $\tuple {e_S, e_T}$ is the identity element for $\struct {S \times T, \circ}$.
Proof
Sufficient Condition
Let $e_S$ and $e_T$ be the identity elements of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ respectively.
| \(\ds \tuple {s, t} \circ \tuple {e_S, e_T}\) | \(=\) | \(\ds \tuple {s \circ_1 e_S, t \circ_2 e_T}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {s, t}\) | ||||||||||||
| \(\ds \tuple {e_S, e_T} \circ \tuple {s, t}\) | \(=\) | \(\ds \tuple {e_S \circ_1 s, e_T \circ_2 t}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {s, t}\) |
Thus $\tuple {e_S, e_T}$ is the identity of $\struct {S \times T, \circ}$.
$\Box$
Necessary Condition
Let $\tuple {e_S, e_T}$ be an identity of $\struct {S \times T, \circ}$.
Then we have:
| \(\ds \forall \tuple {s, t} \in S \times T: \, \) | \(\ds \tuple {s, t} \circ \tuple {e_S, e_T}\) | \(=\) | \(\ds \tuple {s, t}\) | Definition of Identity Element | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \tuple {s \circ_1 e_S, t \circ_2 e_T}\) | \(=\) | \(\ds \tuple {s, t}\) | Definition of External Direct Product | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \forall s \in S, t \in T: \, \) | \(\ds s \circ_1 e_S\) | \(=\) | \(\ds s\) | Equality of Ordered Pairs | |||||||||
| \(\, \ds \land \, \) | \(\ds t \circ_2 e_T\) | \(=\) | \(\ds t\) |
and:
| \(\ds \forall \tuple {s, t} \in S \times T: \, \) | \(\ds \tuple {e_S, e_T} \circ \tuple {s, t}\) | \(=\) | \(\ds \tuple {s, t}\) | Definition of Identity Element | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \tuple {e_S \circ_1 s, e_T \circ_2 t}\) | \(=\) | \(\ds \tuple {s, t}\) | Definition of External Direct Product | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \forall s \in S, t \in T: \, \) | \(\ds e_S \circ_1 s\) | \(=\) | \(\ds s\) | Equality of Ordered Pairs | |||||||||
| \(\, \ds \land \, \) | \(\ds e_T \circ_2 t\) | \(=\) | \(\ds t\) |
Thus $e_S$ and $e_T$ are identity elements of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ respectively.
$\blacksquare$