If we replace the in by an arbitrary element , then we have the set
Before the main definition, we define one more operation: . Some properties: (1) , , , , and ; (2) ; and (3) if , then and .
A lattice is called an orthomodular lattice if
is orthocomplemented, and
(orthomodular law) if , then .
The orthomodular law can be restated as follows: if , then . Equivalently, implies . Note that the equation is automatically true in an arbitrary distributive lattice, even without the assumption that .
For example, the lattice of closed subspaces of a hilbert space is orthomodular. is modular iff is finite dimensional. In addition, if we give the set of (bounded) projection operators on an ordering structure by defining iff , then is lattice isomorphic to , and hence orthomodular.
A simple example of an orthocomplemented lattice that is not orthomodular is the benzene:
Note that , but .
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 of projection operators on a Hilbert space . However, the term was coined by Irving Kaplansky, when it was realized that , while orthocomplemented, is not modular. Rather, it satisfies a variant of the modular law as indicated above.
More generally, an orthomodular poset is an orthocomplemented poset such that
given any pair of orthogonal elements (), their greatest lower bound exists ( exists). Simply put, implies .
for any such that , 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).
|Date of creation||2013-03-22 16:33:06|
|Last modified on||2013-03-22 16:33:06|
|Last modified by||CWoo (3771)|
|Defines||relative orthogonal complement|