Inverse of Relation Isomorphism is Relation Isomorphism
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Theorem
Let $\struct {S, \RR_1}$ and $\struct {T, \RR_2}$ be relational structures.
Let $\phi: \struct {S, \RR_1} \to \struct {T, \RR_2}$ be a bijection.
Then:
- $\phi: \struct {S, \RR_1} \to \struct {T, \RR_2}$
is a relation isomorphism if and only if:
- $\phi^{-1}: \struct {T, \RR_2} \to \struct {S, \RR_1}$
is also a relation isomorphism.
Proof
Follows directly from the definition of relation isomorphism.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.9 \ \text{(b)}$