External Direct Product Inverses/Sufficient Condition
Theorem
Let $\struct {S \times T, \circ}$ be the external direct product of the two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let:
- $s^{-1}$ be an inverse of $s \in \struct {S, \circ_1}$
and:
- $t^{-1}$ be an inverse of $t \in \struct {T, \circ_2}$.
Then $\tuple {s^{-1}, t^{-1} }$ is an inverse of $\tuple {s, t} \in \struct {S \times T, \circ}$.
General Result
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 $\tuple {x_1, x_2, \ldots, x_n} \in S$.
Let $y_k$ be an inverse of $x_k$ in $\struct {S_k, \circ_k}$ for each of $k \in \N^*_n$.
Then $\tuple {y_1, y_2, \ldots, y_n}$ is the inverse of $\tuple {x_1, x_2, \ldots, x_n} \in \SS$ in $\struct {\SS, \circ}$.
Proof
Let:
- $e_S$ be the identity for $\struct {S, \circ_1}$
and:
- $e_T$ be the identity for $\struct {T, \circ_2}$.
Also let:
- $s^{-1}$ be the inverse of $s \in \struct {S, \circ_1}$
and
- $t^{-1}$ be the inverse of $t \in \struct {T, \circ_2}$.
Then:
\(\ds \tuple {s, t} \circ \tuple {s^{-1}, t^{-1} }\) | \(=\) | \(\ds \tuple {s \circ_1 s^{-1}, t \circ_2 t^{-1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {e_S, e_T}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {s^{-1} \circ_1 s, t^{-1} \circ_2 t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {s^{-1}, t^{-1} } \circ \tuple {s, t}\) |
Thus the inverse of $\tuple {s, t}$ is $\tuple {s^{-1}, t^{-1} }$.
$\blacksquare$
Also see
- External Direct Product Associativity
- External Direct Product Commutativity
- External Direct Product Identity
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Theorem $13.1: \ 3^\circ$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.1$