relative complement

A complementPlanetmathPlanetmath of an element in a latticeMathworldPlanetmath 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  dL  is said to be a complement of a relative to I if

ad=c and ad=b.

It is easy to see that ac and ba,  so  aI. Similarly, dI.

An element aL is said to be relatively complemented if for every interval I in L with aI, 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.


  • 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 theoryMathworldPlanetmath: let U be a set and A,B subsets of U, the relative complement of A in B is the set theoretic differencePlanetmathPlanetmath 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