Definition:Order Isomorphism/Definition 2
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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 surjective order embedding.
Then $\phi$ is an order isomorphism.
That is, $\phi$ is an order isomorphism if and only if:
- $(1): \quad \phi$ is surjective
- $(2): \quad \forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$
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
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Definition $5$
- 2002: B.A. Davey and H.A. Priestley: Introduction to Lattices and Order (2nd ed.): Definition $1.4$