Definition:Order Isomorphism/Well-Orderings
Definition
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be well-ordered sets.
Let $\phi: S \to T$ be a bijection such that $\phi: S \to T$ is order-preserving:
- $\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$
Then $\phi$ is an order isomorphism.
Two well-ordered sets $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ are (order) isomorphic if and only if there exists such an order isomorphism between them.
Thus $\struct {S, \preceq_1}$ is described as (order) isomorphic to (or with) $\struct {T, \preceq_2}$, and vice versa.
This may be written $\struct {S, \preceq_1} \cong \struct {T, \preceq_2}$.
Where no confusion is possible, it may be abbreviated to $S \cong T$.
Class-Theoretical Definition
In the context of class theory, the definition follows the same lines:
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
- Order-Preserving Bijection on Wosets is Order Isomorphism, where it is shown that this definition is compatible with that of an order isomorphism between ordered sets.
- 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
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.28$
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals