Complemented Lattice - Orthocomplementation

An orthocomplementation on a bounded lattice is a function that maps each element a to an "orthocomplement" a⊥ in such a way that the following axioms are satisfied:

Complement law

a⊥ ∨ a = 1 and a⊥ ∧ a = 0.

Involution law

a⊥⊥ = a.

Order-reversing

if a ≤ b then b⊥ ≤ a⊥.

An orthocomplemented lattice or ortholattice is a bounded lattice which is equipped with an orthocomplementation. The lattices of subspaces of inner product spaces, and the orthogonal complement operation in these lattices, provide examples of orthocomplemented lattices that are not, in general, distributive.

• Some complemented lattices

• In the pentagon lattice N₅, the node on the right-hand side has two complements.

• The diamond lattice M₃ admits no orthocomplementation.

• The lattice M₄ admits 3 orthocomplementations.

• The hexagon lattice admits a unique orthocomplementation, but it is not uniquely complemented.

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.

Orthocomplemented lattices, like Boolean algebras, satisfy de Morgan's laws:

• (a ∨ b)⊥ = a⊥ ∧ b⊥
• (a ∧ b)⊥ = a⊥ ∨ b⊥.