---
tag: 60FNZ35H6R
kind: concept
name: Strict partial order
msc: [06A06]
sources:
  - url: https://en.wikipedia.org/wiki/Partially_ordered_set
    locator: Strict partial orders
---

## Summary

A strict partial order is a binary relation $<$ that is irreflexive and transitive.
It is the "strictly less than" version of a [[partial order|011C29Q1S1]].
Strict partial orders on a set correspond one to one with partial orders on the same set.
So either one can serve as the basic notion.

## Explanations

### L1 human

A strict partial order says when one element is strictly below another.
No element is strictly below itself, and if $a$ is below $b$ and $b$ is below $c$, then $a$ is below $c$.
Some pairs may be incomparable.
For example, "is a proper subset of" on the subsets of $\set{1, 2}$ puts $\emptyset$ below $\set{1}$, but $\set{1}$ and $\set{2}$ are incomparable.