# Relation Isomorphism Preserves Transitivity

## Theorem

Let $\struct {S, \mathcal R_1}$ and $\struct {T, \mathcal R_2}$ be relational structures.

Let $\struct {S, \mathcal R_1}$ and $\struct {T, \mathcal R_2}$ be (relationally) isomorphic.

Then $\mathcal R_1$ is a transitive relation if and only if $\mathcal R_2$ is a transitive relation.

## Proof

Let $\phi: S \to T$ be a relation isomorphism.

By Inverse of Relation Isomorphism is Relation Isomorphism it follows that $\phi^{-1}: T \to S$ is also a relation isomorphism.

Without loss of generality, it is therefore sufficient to prove only that if $\mathcal R_1$ is transitive, then $\mathcal R_2$ is also transitive.

So, suppose $\mathcal R_1$ is a transitive relation.

Let $y_1, y_2, y_3 \in T$ such that $y_1 \mathrel {\mathcal R_2} y_2$ and $y_2 \mathrel {\mathcal R_2} y_3$.

Let $x_1 = \map {\phi^{-1} } {y_1}$, $x_2 = \map {\phi^{-1} } {y_2}$ and $x_3 = \map {\phi^{-1} } {y_3}$.

As $\phi$ is a bijection it follows from Inverse Element of Bijection that:

- $y_1 = \map \phi {x_1}$
- $y_2 = \map \phi {x_2}$
- $y_3 = \map \phi {x_3}$

As $\phi^{-1}$ is a relation isomorphism it follows that:

- $x_1 = \map {\phi^{-1} } {y_1} \mathrel {\mathcal R_1} \map {\phi^{-1} } {y_2} = x_2$
- $x_2 = \map {\phi^{-1} } {y_2} \mathrel {\mathcal R_1} \map {\phi^{-1} } {y_3} = x_3$

As $\mathcal R_1$ is a transitive relation it follows that:

- $x_1 \mathrel {\mathcal R_1} x_3$

As $\phi$ is a relation isomorphism it follows that:

- $y_1 = \map \phi {x_1} \mathrel {\mathcal R_2} \map \phi {x_3} = y_3$

Hence if $y_1 \mathrel {\mathcal R_2} y_2$ and $y_2 \mathrel {\mathcal R_2} y_3$, then also:

- $y_1 \mathrel {\mathcal R_2} y_3$

Hence, $\mathcal R_2$ is a transitive relation, by definition.

Hence the result.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Exercise $14.9 \ \text{(c)}$