---
tag: Z5E2GX2B81
kind: definition
name: Relatively complemented lattice
msc: [06C15, 06B05]
defines: ES9XR9CQ3D
uses: [WH4SFAACAS, J2T75ATRV5, ZT47BK9Y6K]
sources:
  - url: https://en.wikipedia.org/wiki/Complemented_lattice
---

## Definition

Let $L$ be a [[lattice|WH4SFAACAS]] with [[join and meet|J2T75ATRV5]] $\vee$ and $\wedge$.
For $c \le d$ in $L$, write $[c, d] = \set{x \in L : c \le x \le d}$.
A *relative complement* of $a \in [c, d]$ in $[c, d]$ is an element $b \in [c, d]$ such that
$$a \wedge b = c \quad\text{and}\quad a \vee b = d.$$
The lattice $L$ is *relatively complemented* if for all $c \le d$ in $L$, every element of $[c, d]$ has a relative complement in $[c, d]$.

## Remarks

Equivalently, every interval $[c, d]$ of $L$ is a [[complemented lattice|ZT47BK9Y6K]] with least element $c$ and greatest element $d$.
A relatively complemented lattice with a least and a greatest element is complemented.

Example: the lattice of all subsets of a set $X$.
For $C \subseteq A \subseteq D$, the set $B = C \cup (D \setminus A)$ satisfies $A \cap B = C$ and $A \cup B = D$.

Non-example: the [[pentagon lattice N5|DHC8TZXGYS]] with $0 < a < c < 1$ and $b$ incomparable to $a$ and $c$.
It is complemented, but the interval $[0, c]$ is the chain $0 < a < c$, in which $a$ has no complement.

A complemented [[modular lattice|QX7WZ92QAK]] is relatively complemented: if $x$ is a complement of $a$ and $c \le a \le d$, then $b = c \vee (x \wedge d)$ is a relative complement of $a$ in $[c, d]$.