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.

• 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).

• 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 $B-A$. 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	2013-03-22 15:51:45
Last modified on	2013-03-22 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