---
tag: KRDR04RQ7N
kind: concept
name: Ordered group
msc: ["06F15", "20F60"]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Partially_ordered_group
same_as:
  wikidata: Q2715875
  wikipedia: Partially ordered group
---

## Summary

An ordered group, or partially ordered group, is a [[group|G5S1M9]] with a [[partial order|0QV2JS9RJ3]] that is preserved by multiplication on both sides.
The order is then determined by its positive cone, the set of elements that are at least the identity.
Ordered groups include the additive groups of ordered fields, groups of functions with the pointwise order, and lattice-ordered groups such as groups of real functions.
They are the common setting of the theory of ordered algebraic structures.

## Explanations

### L1 human

In the integers, if $a \le b$ then $a + c \le b + c$: adding the same number to both sides keeps an inequality.
An ordered group is a group with an order that has this property.
The order does not need to compare every two elements.
For example, in the plane one can say $(a, b) \le (c, d)$ when $a \le c$ and $b \le d$.
Then $(1, 0)$ and $(0, 1)$ are not comparable, but adding a vector to both sides still keeps every inequality.