---
tag: 011C29Q1S1
kind: definition
name: Partial order
msc: [06A06]
defines: 0QV2JS9RJ3
uses: [0RXS8R6K37, 01MGYYBB9R]
sources:
  - url: https://en.wikipedia.org/wiki/Partially_ordered_set
    locator: lead section
---

## Definition

Let $A$ be a set.
A partial order on $A$ is a [[preorder|0RXS8R6K37]] $\le$ on $A$ that is [[antisymmetric|01MGYYBB9R]]. For all $a, b, c \in A$:

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

The pair $(A, \le)$ is a partially ordered set.

## Remarks

Examples are $\le$ on $\mathbb{R}$, divisibility on $\mathbb{N}$, and $\subseteq$ on the [[power set|0BC7KX51Z8]] of a set.
"Partial" means that two elements need not be comparable: neither $\{1\} \subseteq \{2\}$ nor $\{2\} \subseteq \{1\}$.
The strict order $a < b$ means $a \le b$ and $a \neq b$.
A finite poset can be drawn as a [[Hasse diagram|00EYM7MDKS]].

## Explanations

### L1 human

A partial order ranks some pairs of elements, but not necessarily all pairs.
In the family tree, "is an ancestor of or equal to" is a partial order: two cousins are not comparable.