Definition:Relation Isomorphism

This page is about Relation Isomorphism in the context of Relation Theory. For other uses, see Isomorphism.

Definition

Let $\struct {S_1, \RR_1}$ and $\struct {S_2, \RR_2}$ be relational structures.

Let there exist a bijection $\phi: S_1 \to S_2$ such that:

$(1): \quad \forall \tuple {s_1, t_1} \in \RR_1: \tuple {\map \phi {s_1}, \map \phi {t_1} } \in \RR_2$

$(2): \quad \forall \tuple {s_2, t_2} \in \RR_2: \tuple {\map {\phi^{-1} } {s_2}, \map {\phi^{-1} } {t_2} } \in \RR_1$

Then $\struct {S_1, \RR_1}$ and $\struct {S_2, \RR_2}$ are isomorphic, and this is denoted $S_1 \cong S_2$.

The function $\phi$ is called a relation isomorphism, or just an isomorphism, from $\struct {S_1, \RR_1}$ to $\struct {S_2, \RR_2}$.

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

Thus isomorphism means equal structure.

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.9 \ \text{(a)}$