External Direct Product Identity/Necessary Condition

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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 $\struct {T, \circ_2}$.


Let $\struct {S \times T, \circ}$ have an identity element $\tuple {e_S, e_T}$.

Then:

$\struct {S, \circ_1}$ has an identity element $e_S$

and:

$\struct {T, \circ_2}$ has an identity element $e_T$.


Proof

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.

$\blacksquare$


Also see


Sources

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