Orthocomplemented lattice
- Orthocomplemented lattice
In lattice theory, a branch of the mathematical discipline called order theory, an orthocomplemented lattice (or just ortholattice) is an algebraic structure consisting of a bounded lattice equipped with an orthocomplementation, i.e. an order-reversing involution that maps each element to its complement. Ortholattices are a natural not necessarily distributive generalization of Boolean algebras.
The orthocomplement of an element "a" is often written as "a"⊥. It satisfies the following axioms.
* Complement law: "a"⊥ ∨ "a" = 1 and "a"⊥ ∧ "a" = 0.
* Involution law: "a"⊥⊥ = "a".
* Order-reversing if "a" ≤ "b" then "b"⊥ ≤ "a"⊥.
The element "a"⊥ is called the orthocomplement of "a".
Properties
Orthocomplemented lattices, like Boolean algebras, satisfy de Morgan's laws:
* ("a" ∨ "b")⊥ = "a"⊥ ∧ "b"⊥
* ("a" ∧ "b")⊥ = "a"⊥ ∨ "b"⊥.
Boolean algebras are a special case of orthocomplemented lattices, which in turn are a special case of complemented lattices (with extra structure). These structures are most often used in quantum logic, where the closed subspaces of a separable Hilbert space represent quantum propositions and behave as an orthocomplemented lattice.
Orthomodular lattices
A lattice is called modular if for all elements "a", "b" and "c" the implication::if "a" ≤ "c", then "a" ∨ ("b" ∧ "c") = ("a" ∨ "b") ∧ "c"holds. This is weaker than distributivity. A natural further weakening of this condition for orthocomplemented lattices, necessary for applications in quantum logic, is to require it only in the special case "b" = "a"⊥. An orthomodular lattice is therefore defined as an orthocomplemented lattice such that for any two elements the implication::if "a" ≤ "c", then "a" ∨ ("a"⊥ ∧ "c") = "c"holds.
Lattices of this form are of crucial importance for the study of quantum logic, since they are part of the axiomisation of the Hilbert space formulation of quantum mechanics.
External links
* [http://planetmath.org/encyclopedia/OrthocomplementedLattice.html Orthocomplemented lattice] at PlanetMath
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