---
tag: 0RXS8R6K37
kind: definition
name: Preorder
msc: [06A06]
defines: 0VABYWJWH9
uses: [0XEGNW851S, 0CYRECG2YQ]
sources:
  - url: https://en.wikipedia.org/wiki/Preorder
    locator: lead section
---

## Definition

Let $A$ be a set.
A preorder on $A$ is a [[binary relation|08EJC6Q25H]] $\le$ on $A$ that is [[reflexive|0XEGNW851S]] and [[transitive|0CYRECG2YQ]]. For all $a, b, c \in A$:

1. $a \le a$;
2. $a \le b$ and $b \le c$ imply $a \le c$.

## Remarks

Divisibility on $\mathbb{Z}$ is a preorder. It is not antisymmetric, because $2 \mid -2$ and $-2 \mid 2$.
An antisymmetric preorder is a [[partial order|011C29Q1S1]].
For a preorder, $a \le b$ and $b \le a$ together define an [[equivalence relation|0SGTMBEWRF]]. The preorder gives a partial order on the [[quotient set|0ZVHXDSZC7]].

## Explanations

### L1 human

Compare people by height with "is at most as tall as".
This is a preorder: two different people can have the same height.