---
tag: ZT47BK9Y6K
kind: concept
name: Complemented lattice
msc: [06C15, 06B05]
specializes: [72V1Z82M5F]
sources:
  - url: https://en.wikipedia.org/wiki/Complemented_lattice
same_as:
  wikidata: Q5156434
  wikipedia: Complemented lattice
---

## Summary

A complemented lattice is a [[bounded lattice|72V1Z82M5F]] in which every element $a$ has a complement: an element $b$ with $a \wedge b = 0$ and $a \vee b = 1$.
The lattice of subspaces of a vector space and every [[Boolean algebra|0XADJKM699]] are complemented.
Complements need not be unique; in a distributive lattice they are, and a complemented distributive lattice is the same as a Boolean algebra.
Complemented modular lattices are the setting of projective geometry and of von Neumann's [[continuous geometry|5CET1KWWWH]].

## Explanations

### L1 human

A complement of $a$ is something that has nothing in common with $a$ but fills up everything together with $a$.
For a line through the origin in the plane, any other line through the origin is a complement.
So a complement exists, but there are many choices.
In the lattice of subsets of a set, the only complement of a subset is the set of points outside it.