Definition:Order Isomorphism/Well-Orderings/Class Theory

Definition

Let $\struct {A, \preccurlyeq_1}$ and $\struct {B, \preccurlyeq_2}$ be well-ordered classes.

Let $\phi: A \to B$ be a bijection such that $\phi: A \to B$ is order-preserving:

$\forall x, y \in S: x \preccurlyeq_1 y \implies \map \phi x \preccurlyeq_2 \map \phi y$

Then $\phi$ is an order isomorphism.

Also see

• Results about order 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.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 2$ Isomorphisms of well orderings