---
tag: P33CMYHJW9
kind: concept
name: Strict total order
msc: [06A05]
specializes: [60FNZ35H6R]
sources:
  - url: https://en.wikipedia.org/wiki/Total_order
    locator: Strict and non-strict total orders
---

## Summary

A strict total order is a [[strict partial order|SWB1ZXFVKY]] in which any two distinct elements are comparable.
It is the strict form of a [[total order|0316AYH1SC]], in the same way that $<$ is the strict form of $\le$ on the real numbers.
For any two elements $a$ and $b$, exactly one of $a < b$, $a = b$, $b < a$ holds.

## Explanations

### L1 human

A strict total order puts all elements on one line and says which of two different elements comes first.
The relation $<$ on the integers is the standard example: for $3$ and $5$, one has $3 < 5$.
Alphabetical order of distinct words is another example.