External Direct Product Inverses/Necessary Condition
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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 $\tuple {s^{-1}, t^{-1} }$ be an inverse of $\tuple {s, t} \in \struct {S \times T, \circ}$.
Then:
- $s^{-1}$ is an inverse of $s \in \struct {S, \circ_1}$
and:
- $t^{-1}$ is an inverse of $t \in \struct {T, \circ_2}$.
Proof
Let $\tuple {e_S, e_T}$ be the identity element of $\struct {S \times T, \circ}$.
Let $\tuple {s^{-1}, t^{-1} }$ be an inverse element of $\tuple {s, t} \in \struct {S \times T, \circ}$.
Then we have:
\(\ds \tuple {s, t} \circ \tuple {s^{-1}, t^{-1} }\) | \(=\) | \(\ds \tuple {e_S, e_T}\) | Definition of Inverse Element | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {s \circ_1 s^{-1}, t \circ_2 t^{-1} }\) | \(=\) | \(\ds \tuple {e_S, e_T}\) | Definition of External Direct Product | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall s \in S, t \in T: \, \) | \(\ds s \circ_1 s^{-1}\) | \(=\) | \(\ds e_S\) | Equality of Ordered Pairs | |||||||||
\(\, \ds \land \, \) | \(\ds t \circ_2 t^{-1}\) | \(=\) | \(\ds e_T\) |
and:
\(\ds \tuple {s^{-1}, t^{-1} } \circ \tuple {s, t}\) | \(=\) | \(\ds \tuple {e_S, e_T}\) | Definition of Inverse Element | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {s^{-1} \circ_1 s, t^{-1} \circ_2 t}\) | \(=\) | \(\ds \tuple {e_S, e_T}\) | Definition of External Direct Product | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds s^{-1} \circ_1 s\) | \(=\) | \(\ds e_S\) | Equality of Ordered Pairs | ||||||||||
\(\, \ds \land \, \) | \(\ds t^{-1} \circ_2 t\) | \(=\) | \(\ds e_T\) |
Thus $s^{-1}$ and $t^{-1}$ are inverse elements of $s \in \struct {S, \circ_1}$ and $t \in \struct {T, \circ_2}$ respectively.
$\blacksquare$
Also see
- External Direct Product Associativity
- External Direct Product Commutativity
- External Direct Product Inverses
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: Exercise $13.5$