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 be a lattice, an element of , and an interval (http://planetmath.org/LatticeInterval) in . An element is said to be a complement of relative to if
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.
|Date of creation||2013-03-22 15:51:45|
|Last modified on||2013-03-22 15:51:45|
|Last modified by||CWoo (3771)|
|Defines||relatively complemented lattice|