Definition:Order Isomorphism/Definition 3
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Definition
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\phi: S \to T$ be a bijection such that:
- $\forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$
Then $\phi$ is an order isomorphism.
Also see
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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.2$: Order-preserving mappings. Isomorphisms