Definition:Order Isomorphism
This page is about Isomorphism in the context of Order Theory. For other uses, see Isomorphism.
Definition
Definition 1
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.
Definition 2
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ 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
- $(2): \quad \forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$
Two ordered sets $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ are (order) isomorphic if there exists such an order isomorphism between them.
$\struct {S, \preceq_1}$ 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 $S \cong T$.
Well-Ordered Sets
When $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ are well-ordered sets, the condition on the order preservation can be relaxed:
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.
Also see
- Equivalence of Definitions of Order Isomorphism
- Definition:Relation Isomorphism, from which it can be seen that order isomorphism is a special case.
- Inverse of Increasing Bijection need not be Increasing
- 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: $\S 6.28$
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S \text I.2$