---
tag: 0Y74KN99A5
kind: definition
name: Order isomorphism
msc: [06A06]
defines: 0D3WA36K2Q
uses: [0SNQEF7Z53, 011C29Q1S1]
sources:
  - url: https://en.wikipedia.org/wiki/Order_isomorphism
    locator: lead section
---

## Definition

Let $(P, \le_P)$ and $(Q, \le_Q)$ be [[partially ordered sets|011C29Q1S1]].
An order isomorphism from $P$ to $Q$ is a [[bijective function|0SNQEF7Z53]] $f \colon P \to Q$ such that, for all $a, b \in P$,
$$a \le_P b \iff f(a) \le_Q f(b).$$

## Remarks

The inverse of an order isomorphism is an order isomorphism.
The map $x \mapsto 2x$ is an order isomorphism from $\mathbb{Z}$ to the even integers.
The ordered sets $\mathbb{N}$ and $\mathbb{Z}$ are not order isomorphic, because $\mathbb{N}$ has a least element and $\mathbb{Z}$ has none.
A [[monotone|0WFF6AZPBP]] bijection need not be an order isomorphism, when the orders are not total.

## Explanations

### L1 human

Order isomorphic sets have the same shape as ordered sets.
Only the names of the elements differ.