External Direct Product Identity

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 $\struct {S \times T, \circ}$.
 * $\tuple {e_S, e_T}$ is the identity element for $\struct {S \times T, \circ}$.

Also see

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