---
tag: P2E8BNYC7B
kind: concept
name: Linearly ordered group
msc: ["06F15", "20F60"]
specializes: [KRDR04RQ7N]
sources:
  - url: https://en.wikipedia.org/wiki/Linearly_ordered_group
same_as:
  wikidata: Q10292020
  wikipedia: Linearly ordered group
---

## Summary

A linearly ordered group is an [[ordered group|KRDR04RQ7N]] whose order is a [[total order|0FRXBVVR03]].
Any two of its elements are comparable, and the order is preserved by multiplication on both sides.
The additive groups of $\Z$, $\Q$ and $\R$ are the basic examples.
Linearly ordered abelian groups are the value groups of valuations, and they are described by Hölder's theorem and the Hahn embedding theorem.

## Explanations

### L1 human

A linearly ordered group is a group whose elements stand in one line, from smaller to larger.
Multiplying by a fixed element on either side shifts the whole line without changing the order of its points.
The integers under addition are an example: adding $5$ to every integer keeps them in the same order.
The pairs of integers with the dictionary order are another example, where $(0, 1000) < (1, 0)$.