External Direct Product Identity/Necessary Condition

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:

and:

Thus $e_S$ and $e_T$ are identity elements of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ respectively.

Also see

 * External Direct Product Associativity
 * External Direct Product Commutativity
 * External Direct Product Inverses