orthomodular lattice

Orthogonality Relations

Let L be an orthocomplemented lattice and a,bL. a is said to be orthogonalMathworldPlanetmathPlanetmath to b if ab, denoted by ab. If ab, then b=ba, so is a symmetric relationMathworldPlanetmath on L. It is easy to see that, for any a,bL, ab implies ab=0, and aa.

For any aL, define M(a):={cLca and 1=ca}. An element of M(a) is called an orthogonal complementMathworldPlanetmath of a. We have aM(a), and any orthogonal complement of a is a complementPlanetmathPlanetmath of a.

If we replace the 1 in M(a) by an arbitrary element ba, then we have the set

M(a,b):={cLca and b=ca}.

An element of M(a,b) is called an orthogonal complement of a relative to b. Clearly, M(a)=M(a,1). Also, for a,cb, cM(a,b) iff aM(c,b). As a result, we can define a symmetricPlanetmathPlanetmathPlanetmath binary operator on [0,b], given by b=ac iff cM(a,b). Note that b=b0.

Before the main definition, we define one more operationMathworldPlanetmath: b-a:=ba. Some properties: (1) a-a=0, a-0=a, 0-a=0, a-1=0, and 1-a=a; (2) b-a=a-b; and (3) if ab, then a(b-a) and a(b-a)b.


A latticeMathworldPlanetmath L is called an orthomodular lattice if

  1. 1.

    L is orthocomplemented, and

  2. 2.

    (orthomodular law) if xy, then y=x(y-x).

The orthomodular law can be restated as follows: if xy, then y=x(yx). Equivalently, xy implies y=(yx)(yx). Note that the equation is automatically true in an arbitrary distributive latticeMathworldPlanetmath, even without the assumptionPlanetmathPlanetmath that xy.

For example, the lattice (H) of closed subspaces of a hilbert space H is orthomodular. (H) is modular iff H is finite dimensional. In addition, if we give the set (H) of (bounded) projection operators on H an ordering structureMathworldPlanetmath by defining PQ iff P(H)Q(H), then (H) is lattice isomorphic to (H), and hence orthomodular.

A simple example of an orthocomplemented lattice that is not orthomodular is the benzene:


Note that ab, but a(ba)=a0=ab.

An nice example of an orthomodular lattice that is not modular can be found in the reference below.


  • Orthomodular lattices were first studied by John von Neumann and Garett Birkhoff, when they were trying to develop the logic of quantum mechanics (http://planetmath.org/QuantumLogic) by studying the structure of the lattice (H) of projection operators on a Hilbert space H. However, the term was coined by Irving Kaplansky, when it was realized that (H), while orthocomplemented, is not modular. Rather, it satisfies a variant of the modular law as indicated above.

  • More generally, an orthomodular poset P is an orthocomplemented poset such that

    1. (a)

      given any pair of orthogonal elements x,yP (xy), their greatest lower boundMathworldPlanetmath exists (xy exists). Simply put, xy implies xyP.

    2. (b)

      for any x,yP such that xy, the orthomodular law holds (the right hand side of the orthomodular law exists via the first condition).

    From this definition, we see that an orthomodular lattice is just an orthomodular poset that is also a lattice.


  • 1 L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).
Title orthomodular lattice
Canonical name OrthomodularLattice
Date of creation 2013-03-22 16:33:06
Last modified on 2013-03-22 16:33:06
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 06C15
Classification msc 81P10
Classification msc 03G12
Related topic OrthocomplementedLattice
Related topic LatticeOfProjections
Defines orthomodular poset
Defines orthogonal
Defines orthogonal complement
Defines relative orthogonal complement