relative complement
A complement^{} of an element in a lattice^{} is only defined when the lattice in question is bounded (http://planetmath.org/BoundedLattice). In general, a lattice is not bounded and there are no complements to speak of. Nevertheless, if the sublattice of a lattice is bounded, we can speak of complements of an element relative to that sublattice.
Let $L$ be a lattice, $a$ an element of $L$, and $I=[b,c]$ an interval (http://planetmath.org/LatticeInterval) in $L$. An element $d\in L$ is said to be a complement of $a$ relative to $I$ if
$$a\vee d=c\text{and}a\wedge d=b.$$ 
It is easy to see that $a\le c$ and $b\le a$, so $a\in I$. Similarly, $d\in I$.
An element $a\in L$ is said to be relatively complemented if for every interval $I$ in $L$ with $a\in I$, it has a complement relative to $I$. The lattice $L$ itself is called a relatively complemented lattice if every element of $L$ is relatively complemented. Equivalently, $L$ is relatively complemented iff each of its interval is a complemented lattice.
Remarks.

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A relatively complemented lattice is complemented if it is bounded. Conversely, a complemented lattice is relatively complemented if it is modular (http://planetmath.org/ModularLattice).

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The notion of a relative complement of an element in a lattice has nothing to do with that found in set theory^{}: let $U$ be a set and $A,B$ subsets of $U$, the relative complement of $A$ in $B$ is the set theoretic difference^{} $BA$. While the relative difference is necessarily a subset of $B$, $A$ does not have to be a subset of $B$.
Title  relative complement 

Canonical name  RelativeComplement 
Date of creation  20130322 15:51:45 
Last modified on  20130322 15:51:45 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06C15 
Related topic  RelativePseudocomplement 
Related topic  BrouwerianLattice 
Defines  relatively complemented lattice 
Defines  relatively complemented 