---
tag: NT44831EFE
kind: concept
name: Successor order
msc: [06A06, 06A05]
sources:
  - url: https://en.wikipedia.org/wiki/Successor_function
---

## Summary

A successor order is a partial order with a successor map.
This map sends each non-maximal element to the least element strictly above it.
It abstracts the [[successor function|0AK9TGSNXY]] $n \mapsto n + 1$ on the natural numbers.
The natural numbers, the integers, and every well-ordered set carry a successor order, while the rationals do not.

## Explanations

### L1 human

In the integers every number has a next number: after $n$ comes $n + 1$, and nothing lies strictly between them.
A successor order is an order with such a "next element" map.
In the rationals there is no next number after $0$, since between $0$ and any positive rational there is another rational.
So the rationals have no successor order.