---
tag: C9GDH7264N
kind: definition
name: Strict total order
msc: [06A05]
defines: P33CMYHJW9
uses: [SWB1ZXFVKY]
sources:
  - url: https://en.wikipedia.org/wiki/Total_order
    locator: Strict and non-strict total orders
---

## Definition

Let $A$ be a set.
A *strict total order* on $A$ is a [[strict partial order|SWB1ZXFVKY]] $<$ on $A$ such that, for all $a, b \in A$ with $a \ne b$, $a < b$ or $b < a$.

## Remarks

The relation $<$ on $\R$ is a strict total order: it is irreflexive, transitive, and for distinct reals $x$ and $y$, $x < y$ or $y < x$.
The relation $\subsetneq$ on the [[power set|0BC7KX51Z8]] of $\set{1, 2}$ is a strict partial order but not a strict total order.
The sets $\set{1}$ and $\set{2}$ are distinct, and neither is a proper subset of the other.
The condition gives trichotomy: for all $a, b \in A$, exactly one of $a < b$, $a = b$, $b < a$ holds.
The strict total orders on $A$ correspond one to one with the [[total orders|0316AYH1SC]] on $A$, by $a \le b$ if and only if $a < b$ or $a = b$.
Other names are strict linear order and strict simple order.

## Explanations

### L1 human

A strict total order ranks all elements in one line with no ties.
Given two different elements, it tells which one comes first.