complemented lattice

Let L be a bounded latticeMathworldPlanetmath (with 0 and 1), and aL. A complementPlanetmathPlanetmath of a is an element bL such that

ab=0 and ab=1.

Remark. Complements may not exist. If L is a non-trivial chain, then no element (other than 0 and 1) has a complement. This also shows that if a is a complement of a non-trivial element b, then a and b form an antichainMathworldPlanetmath.

An element in a bounded lattice is complemented if it has a complement. A complemented lattice is a bounded lattice in which every element is complemented.


  • In a complemented lattice, there may be more than one complement corresponding to each element. Two elements are said to be related, or perspective if they have a common complement. For example, the following latticeMathworldPlanetmath is complemented.


    Note that none of the non-trivial elements have unique complements. Any two non-trivial elements are related via the third.

  • If a complemented lattice L is a distributive latticeMathworldPlanetmath, then L is uniquely complemented (in fact, a Boolean lattice). For if y1 and y2 are two complements of x, then


    Similarly, y1=y2y1. So y2=y1.

  • In the category of complemented lattices, a morphism between two objects is a {0,1}-lattice homomorphismMathworldPlanetmath; that is, a lattice homomorphism which preserves 0 and 1.

Title complemented lattice
Canonical name ComplementedLattice
Date of creation 2013-03-22 15:02:25
Last modified on 2013-03-22 15:02:25
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 26
Author CWoo (3771)
Entry type Definition
Classification msc 06C15
Classification msc 06B05
Synonym perspective elements
Synonym complemented
Related topic Perspectivity
Related topic OrthocomplementedLattice
Related topic PseudocomplementedLattice
Related topic DifferenceOfLatticeElements
Related topic Pseudocomplement
Defines related elements in lattice
Defines complement
Defines complemented element