relative complementRelativeComplement2006-04-21 19:12:182008-04-08 01:46:26Definitioncomplemented latticerelatively complemented latticerelatively complemented\usepackage{amssymb,amscd}
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A complement of an element in a lattice is only defined when the lattice in question is \PMlinkname{bounded}{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 \emph{relative} to that sublattice.
Let $L$ be a lattice, $a$ an element of $L$, and $I=[b,c]$ an \PMlinkname{interval}{LatticeInterval} in $L$. An element\, $d\in L$\, is said to be a complement of $a$ \emph{relative} to $I$ if
$$a\vee d=c\,\mbox{ and }\,a\wedge d=b.$$
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 \emph{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 \emph{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.
\textbf{Remarks}.
\begin{itemize}
\item A relatively complemented lattice is complemented if it is bounded. Conversely, a complemented lattice is relatively complemented if it is
\PMlinkname{modular}{ModularLattice}.
\item 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 $B$.
\end{itemize}