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.