Orthocomplemented lattice

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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.

[edit] 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.

[edit] 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.