---
tag: MRYFDAG5G8
kind: concept
name: Duality in order theory
msc: [06A06]
sources:
  - url: https://en.wikipedia.org/wiki/Duality_(order_theory)
same_as:
  wikidata: Q554403
  wikipedia: Duality (order theory)
---

## Summary

Every [[partially ordered set|011C29Q1S1]] has a dual, obtained by reversing the order.
Each order-theoretic notion has a dual notion: upper bound and lower bound, supremum and infimum, greatest and least element.
The duality principle says that a statement true for all posets stays true when every notion in it is replaced by its dual.
This halves the work in many proofs.

## Explanations

### L1 human

Turn a picture of an order upside down, and you get another order: its dual.
What was the top is now the bottom, and what was above is now below.
So every theorem about orders comes with a mirror theorem for free.
For example, the divisors of $12$ ordered by divisibility have greatest element $12$; in the dual order, $12$ is the least element.