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 be a lattice, an element of , and an interval in . An element is said to be a complement of relative to if

and

It is easy to see that and , so . Similarly, .

An element is said to be relatively complemented if for every interval in with , it has a complement relative to . The lattice itself is called a relatively complemented lattice if every element of is relatively complemented. Equivalently, 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 be a set and subsets of , the relative complement of in is the set theoretic difference . While the relative difference is necessarily a subset of , does not have to be a subset of .