commutativity relation in an orthocomplemented lattice

Let L be an orthocomplemented lattice with a,bL. We say that a commutes with b if a=(ab)(ab). When a commutes with b, we write aCb. Dualize everything, we have that a dually commutes with b, written aDb, if a=(ab)(ab).

Some properties. Below are some properties of the commutativity relationsMathworldPlanetmath C and D.

  1. 1.

    C is reflexiveMathworldPlanetmathPlanetmath.

  2. 2.

    aCb iff aCb.

  3. 3.

    aCb iff aDb.

  4. 4.

    if ab or ab, then aCb.

  5. 5.

    a is said to orthogonally commute with b if aCb and bCa. In this case, we write aMb. The terminology comes from the following fact: aMb iff there are x,y,z,tL, with:

    1. (a)

      xy (x is orthogonal to y, or xy),

    2. (b)


    3. (c)


    4. (d)

      a=xy, and

    5. (e)


  6. 6.

    C is symmetric iff D=C(=M) iff L is an orthomodular lattice.

  7. 7.

    C is an equivalence relationMathworldPlanetmath iff C=L×L iff L is a Boolean algebraMathworldPlanetmath.

Remark. More generally, one can define commutativity C on an orthomodular poset P: for a,bP, aCb iff ab, ab, and (ab)(ab) exist, and (ab)(ab)=a. Dual commutativity and mutual commutativity in an orthomodular poset are defined similarly (with the provision that the binary operationsMathworldPlanetmath on the pair of elements are meaningful).


  • 1 L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).
Title commutativity relation in an orthocomplemented lattice
Canonical name CommutativityRelationInAnOrthocomplementedLattice
Date of creation 2013-03-22 16:43:22
Last modified on 2013-03-22 16:43:22
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 06C15
Classification msc 03G12
Defines dually commute
Defines orthogonally commute