uniquely complemented lattice

Recall that in a bounded distributive latticeMathworldPlanetmath, complementsMathworldPlanetmathPlanetmath, relative complements, and differences of lattice elements, if exist, must be unique. This leads to the general consideration of general bounded lattices in which complements are unique.

Definition. A complemented lattice such that every element has a unique complement is said to be uniquely complemented. If a is an element of a uniquely complemented lattice, a denotes its (unique) complement. One can think of as a unary operator on the latticeMathworldPlanetmath.

One of the first consequences is


To see this, we have that aa=1, aa=0, as well as a′′a=1, a′′a=0. So a=a′′, since they are both complements of a.

Below are some additional (and non-trivial) properties of a uniquely complemented lattice:

  • there exists a uniquely complemented lattice that is not distributivePlanetmathPlanetmath

  • a uniquely complemented lattice L is distributive if at least one of the following is satisfied:

    1. (a)

      , as an operator on L, is order reversing;

    2. (b)


    3. (c)


    4. (d)

      (von Neumann) L is a modular latticeMathworldPlanetmath;

    5. (e)

      (Birkhoff-Ward) L is an atomic lattice.

    In fact, the first three conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath, so that L is distributive if it satisfies the de Morgan’s laws.

  • (Dilworth) every lattice can be embedded in a uniquely complemented lattice.


Title uniquely complemented lattice
Canonical name UniquelyComplementedLattice
Date of creation 2013-03-22 17:58:15
Last modified on 2013-03-22 17:58:15
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 06B05
Classification msc 06C15
Defines uniquely complemented