---
tag: RW7YC3HG0S
kind: concept
name: Jordan–Dedekind chain condition
msc: [06A07, 06B05]
sources:
  - url: https://en.wikipedia.org/wiki/Graded_poset
---

## Summary

A partially ordered set satisfies the Jordan–Dedekind chain condition when, between any two comparable elements, all [[maximal chains|3AX549FFFM]] have the same finite length.
It is the property behind the Jordan–Hölder theorem, which says that composition series of a group have the same length.
For a finite poset with a least element, it is the same as being a [[graded poset|1C7TKGTB2F]].

## Explanations

### L1 human

Think of climbing from one element to a higher one by steps that cannot be split further.
The condition says that every such climb takes the same number of steps, whatever route is chosen.
In the divisors of $12$, every climb from $1$ to $12$ by single prime factors takes $3$ steps, such as $1, 2, 4, 12$ or $1, 3, 6, 12$.