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orthomodular lattice
Orthogonality Relations
Let be an orthocomplemented lattice and . is said to be orthogonal to if , denoted by . If , then , so is a symmetric relation on . It is easy to see that, for any , implies , and .
For any , define . An element of is called an orthogonal complement of . We have , and any orthogonal complement of is a complement of .
If we replace the in by an arbitrary element , then we have the set
An element of is called an orthogonal complement of relative to . Clearly, . Also, for , iff . As a result, we can define a symmetric binary operator on , given by iff . Note that .
Before the main definition, we define one more operation: . Some properties: (1) , , , , and ; (2) ; and (3) if , then and .
Definition
A lattice is called an orthomodular lattice if
1. is orthocomplemented, and
2. (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.
Note that , but .
An nice example of an orthomodular lattice that is not modular can be found in the reference below.
Remarks.
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Orthomodular lattices were first studied by John von Neumann and Garett Birkhoff, when they were trying to develop the logic of quantum mechanics 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.
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More generally, an orthomodular poset is an orthocomplemented poset such that
(a) given any pair of orthogonal elements (), their greatest lower bound exists ( exists). Simply put, implies .
(b) 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.
References
- 1 L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).
Mathematics Subject Classification
06C15 Complemented lattices, orthocomplemented lattices and posets81P10 Logical foundations of quantum mechanics; quantum logic
03G12 Quantum logic
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