Definition:Relation Isomorphism

Definition
Let $\left({S_1, \mathcal R_1}\right)$ and $\left({S_2, \mathcal R_2}\right)$ be relational structures.

Let there exist a bijection $\phi: S_1 \to S_2$ such that:
 * $\forall \left({s_1, t_1}\right) \in \mathcal R_1: \left({\phi \left({s_1}\right), \phi \left({t_1}\right)}\right) \in \mathcal R_2$;
 * $\forall \left({s_2, t_2}\right) \in \mathcal R_2: \left({\phi^{-1} \left({s_2}\right), \phi^{-1} \left({t_2}\right)}\right) \in \mathcal R_1$.

Then $\left({S_1, \mathcal R_1}\right)$ and $\left({S_2, \mathcal R_2}\right)$ are isomorphic, and this is denoted $S_1 \cong S_2$.

The function $\phi$ is called a relation isomorphism, or just an isomorphism, from $\left({S_1, \mathcal R_1}\right)$ to $\left({S_2, \mathcal R_2}\right)$.

It follows from this definition that Relation Isomorphism is an Equivalence.