---
tag: 839P1ZNRD7
kind: concept
name: Down-closure
msc: [06A06]
sources:
  - url: https://en.wikipedia.org/wiki/Upper_and_lower_sets
---

## Summary

The down-closure of a subset $S$ of a [[partially ordered set|011C29Q1S1]] is the set of all elements that lie below some element of $S$.
It is the smallest [[lower set|BWYFKCWRW5]] that contains $S$.
The down-closure of a single element $x$ is the principal down-set $\downarrow x$ of all elements below $x$.

## Explanations

### L1 human

To take the down-closure of a set, add everything that lies below one of its elements.
In the real numbers, the down-closure of $\set{1, 5}$ is the half-line $(-\infty, 5]$.
In the positive integers ordered by divisibility, the down-closure of $\set{12}$ is the set of divisors of $12$.
The result is closed downwards, and it is the least such set that contains the original one.