---
tag: ES34ZH23RS
kind: concept
name: Order theory
msc: [06A06, 06B05]
sources:
  - url: https://en.wikipedia.org/wiki/Order_theory
same_as:
  wikidata: Q24175351
  wikipedia: Order theory
---

## Summary

Order theory is the branch of mathematics that studies orders, that is, binary relations that compare elements, such as [[partial orders|011C29Q1S1]], [[total orders|0316AYH1SC]] and [[preorders|0RXS8R6K37]].
Its basic objects are partially ordered sets and the [[monotone functions|0WFF6AZPBP]] between them.
It includes the theory of [[lattices|WH4SFAACAS]], [[Galois connections|MBVJ4JQXC5]] and [[closure operators|17FVAN6HXX]].
Its ideas appear in algebra, topology, logic and computer science, wherever objects can be compared by size, inclusion or strength.

## Explanations

### L1 human

Many collections in mathematics come with a natural way to say that one element is "below" another.
Numbers are compared by size, sets by inclusion, and whole numbers by divisibility.
Order theory studies what all these comparisons have in common.
A picture of a finite order is a [[Hasse diagram|00EYM7MDKS]]: dots for the elements and upward lines for the steps between them.
For example, the divisors $1, 2, 3, 6$ of $6$ form a diamond: $1$ at the bottom, $2$ and $3$ in the middle, $6$ at the top.