Definition:Order Isomorphism/Definition 1
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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:
- $\phi: S \to T$ 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
- $\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$
- $\forall p, q \in T: p \preceq_2 q \implies \map {\phi^{-1} } p \preceq_1 \map {\phi^{-1} } q$
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
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S \text I.2$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations