---
tag: Z15FXM9WM1
kind: concept
name: Order filter
msc: [06A06, 06B10]
sources:
  - url: https://en.wikipedia.org/wiki/Filter_(mathematics)
same_as:
  wikidata: Q1052692
---

## Summary

A filter of a [[partially ordered set|0QV2JS9RJ3]] is a nonempty subset that is closed upward.
Moreover, any two of its elements have a common lower bound inside it.
In a [[lattice|WH4SFAACAS]] this means: closed upward and closed under meets.
Filters are the order duals of [[lattice ideals|5GW2T2AM1A]].
A [[filter on a set|15PF73BKY6]] $X$ is a proper order filter in the lattice of subsets of $X$.

## Explanations

### L1 human

An order filter is a collection of "large" elements.
Anything above a large element is large, and two large elements always have a large element below both of them.
For a fixed element $a$, the set of all elements above $a$ is a filter.
In the subsets of the natural numbers, the sets with finite complement form a filter, since the intersection of two such sets again has finite complement.