sectionally complemented lattice

Proposition 1.

Let L be a latticeMathworldPlanetmath with the least element 0. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    Every pair of elements have a difference (

  2. 2.

    for any aL, the lattice interval [0,a] is a complemented latticeMathworldPlanetmath.


Suppose first that every pair of elements have a difference. Let b[0,a] and let c be a difference between a and b. So 0=bc and cb=ba=a, since ba. This shows that c is a complement of b in [0,a].

Next suppose that [0,a] is complemented for every aL. Let x,yL be any two elements in L. Let a=xy. Since [0,a] is complemented, y has a complement, say z[0,a]. This means that yz=0 and yz=a=xy. Therefore, z is a difference of x and y. ∎

Definition. A lattice L with the least element 0 satisfying either of the two equivalent conditions above is called a sectionally complemented lattice.

Every relatively complemented lattice is sectionally complemented. Every sectionally complemented distributive latticeMathworldPlanetmath is relatively complemented.

Dually, one defines a dually sectionally complemented lattice to be a lattice L with the top element 1 such that for every aL, the interval [a,1] is complemented, or, equivalently, the lattice dual L is sectionally complemented.


  • 1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998)
Title sectionally complemented lattice
Canonical name SectionallyComplementedLattice
Date of creation 2013-03-22 17:58:46
Last modified on 2013-03-22 17:58:46
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 06C15
Classification msc 06B05
Related topic DifferenceOfLatticeElements
Defines sectionally complemented
Defines dually sectionally complemented lattice