Definition:Order Isomorphism

From ProofWiki
Jump to navigation Jump to search

This page is about isomorphisms in order theory. For other uses, see Definition: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 $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ are (order) isomorphic if there exists such an order isomorphism between them.

$\left({S, \preceq_1}\right)$ is described as (order) isomorphic to (or with) $\left({T, \preceq_2}\right)$, and vice versa.

This may be written $\left({S, \preceq_1}\right) \cong \left({T, \preceq_2}\right)$.

Where no confusion is possible, it may be abbreviated to $S \cong T$.


Well-Ordered Sets

When $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ are well-ordered sets, the condition on the order preservation can be relaxed:


Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be well-ordered sets.

Let $\phi: S \to T$ be a bijection such that $\phi: S \to T$ is order-preserving:

$\forall x, y \in S: x \mathop {\preceq_1} y \implies \phi \left({x}\right) \mathop {\preceq_2} \phi \left({y}\right)$


Then $\phi$ is an order isomorphism.


Also see

  • 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