---
tag: T08FF77H16
kind: definition
name: Complemented lattice
msc: [06C15, 06B05]
defines: ZT47BK9Y6K
uses: [72V1Z82M5F, J2T75ATRV5]
sources:
  - url: https://en.wikipedia.org/wiki/Complemented_lattice
---

## Definition

Let $L$ be a [[bounded lattice|72V1Z82M5F]] with least element $0$, greatest element $1$, and [[join and meet|J2T75ATRV5]] $\vee$ and $\wedge$.
A *complement* of an element $a \in L$ is an element $b \in L$ such that
$$a \wedge b = 0 \quad\text{and}\quad a \vee b = 1.$$
The lattice $L$ is *complemented* if every element of $L$ has at least one complement.

## Remarks

Example: the lattice of subspaces of a finite-dimensional vector space $V$.
For a subspace $U$, extend a basis of $U$ to a basis of $V$ and let $W$ be the span of the added vectors; then $U \cap W = \set{0}$ and $U + W = V$.

Non-example: the chain $0 < a < 1$.
The element $a$ has no complement: $a \wedge x = 0$ forces $x = 0$, and then $a \vee x = a \ne 1$.

In the [[diamond lattice M3|JQ6FJH2TP8]] each of the three middle elements has the other two as complements, so complements are not unique.
In a [[distributive lattice|0HM0HQCK6N]] an element has at most one complement, and a complemented distributive lattice is a [[Boolean algebra|0XADJKM699]].