---
tag: JC4J24MF93
kind: concept
name: Order dimension
msc: [06A07]
sources:
  - url: https://en.wikipedia.org/wiki/Order_dimension
same_as:
  wikidata: Q7100431
  wikipedia: Order dimension
---

## Summary

The order dimension of a [[partially ordered set|0QV2JS9RJ3]] is the least number of [[linear extensions|W8DZ5KGGHN]] whose intersection is the given order.
It measures how far the order is from being total: total orders have dimension $1$.
Equivalently, it is the least $d$ such that the poset embeds in a product of $d$ chains with the [[product order|K0AWJK6NZH]].

## Explanations

### L1 human

Imagine ranking a set of objects by several criteria, such as price and quality.
Say that one object beats another when it is better in every criterion.
The dimension of a partial order is the least number of criteria that produce exactly this order.
For points in the plane with $(a, b) \le (c, d)$ when $a \le c$ and $b \le d$, two criteria suffice, so the dimension is at most $2$.