---
tag: ES9XR9CQ3D
kind: concept
name: Relatively complemented lattice
msc: [06C15, 06B05]
specializes: [WH4SFAACAS]
sources:
  - url: https://en.wikipedia.org/wiki/Complemented_lattice
---

## Summary

A relatively complemented lattice is a [[lattice|WH4SFAACAS]] in which every interval $[c, d]$ is a [[complemented lattice|ZT47BK9Y6K]].
So inside every interval each element has a complement relative to the ends of the interval.
A bounded relatively complemented lattice is complemented, and a complemented [[modular lattice|QX7WZ92QAK]] is relatively complemented.
Boolean algebras and lattices of subspaces of a vector space are examples.

## Explanations

### L1 human

Being complemented is a statement about the whole lattice, from its bottom to its top.
Relative complements ask the same thing for every smaller window $[c, d]$ of the lattice.
For subspaces $C \subseteq A \subseteq D$ of a vector space, there is a subspace $B$ between $C$ and $D$ with $A \cap B = C$ and $A + B = D$.
So subspace lattices are complemented not only globally but in every window.