Definition:Order Isomorphism/Definition 1
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Definition
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.
Let $\phi: S \to T$ be a bijection such that:
- $\phi: S \to T$ is order-preserving
- $\phi^{-1}: T \to S$ is order-preserving.
Then $\phi$ is an order isomorphism.
That is, $\phi$ is an order isomorphism if and only if:
- $\phi$ is bijective
- $\forall x, y \in S: x \preceq_1 y \implies \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$
- $\forall p, q \in T: p \preceq_2 q \implies \phi^{-1} \left({p}\right) \preceq_1 \phi^{-1} \left({q}\right)$
So an order isomorphism can be described as a bijection that preserves ordering in both directions.
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): $\S 14$
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.2$: Order-preserving mappings. Isomorphisms
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations