relative complement

A complement of an element in a lattice is only defined when the lattice in question is bounded. 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 in $L$ . An element $d\in L$ is said to be a complement of $a$ 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 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.
• 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 $B$ .