---
tag: D7A0ZGZT5S
kind: concept
name: Complete lattice
msc: [06B23]
specializes: [WH4SFAACAS]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_lattice
same_as:
  wikidata: Q2362924
  wikipedia: Complete lattice
---

## Summary

A complete lattice is a partially ordered set in which every subset, finite or infinite, has a supremum and an infimum.
It is the setting of many fixed-point theorems, such as the [[Knaster–Tarski theorem|RHXP3HA64M]].
Power sets, lattices of subgroups and lattices of closed sets of a topological space are complete lattices.
Every partially ordered set embeds in a complete lattice, its [[Dedekind–MacNeille completion|DKH4933HMQ]].

## Explanations

### L1 human

In a lattice, any two elements have a least upper bound and a greatest lower bound.
A complete lattice asks the same for every collection of elements, even an infinite one.
The subsets of a set, ordered by inclusion, form a complete lattice: the least upper bound of any family is its union and the greatest lower bound is its intersection.
The real line is not complete in this sense, since the set of all integers has no upper bound, but the closed interval $[0, 1]$ is.