---
tag: 0316AYH1SC
kind: definition
name: Total order
msc: [06A05]
defines: 0FRXBVVR03
uses: [011C29Q1S1]
sources:
  - url: https://en.wikipedia.org/wiki/Total_order
    locator: lead section
---

## Definition

Let $A$ be a set.
A total order on $A$ is a [[partial order|011C29Q1S1]] $\le$ on $A$ such that, for all $a, b \in A$, $a \le b$ or $b \le a$.

## Remarks

Other names are linear order and simple order.
The usual orders on $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ are total orders.
The order $\subseteq$ on the power set of $\{1, 2\}$ is not total.

## Explanations

### L1 human

In a total order, all elements stand on one line.
For any two of them, one comes before the other, or they are equal.