# 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:

$(1): \quad \forall \left({s_1, t_1}\right) \in \mathcal R_1: \left({\phi \left({s_1}\right), \phi \left({t_1}\right)}\right) \in \mathcal R_2$
$(2): \quad \forall \left({s_2, t_2}\right) \in \mathcal R_2: \left({\phi^{-1} \left({s_2}\right), \phi^{-1} \left({t_2}\right)}\right) \in \mathcal R_1$

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 $\left({S_2, \mathcal R_2}\right)$.

## Also see

• Results about relation isomorphisms can be found here.

## Linguistic Note

The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.