Complemented lattice

Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra.

[6] Boolean algebras are a special case of orthocomplemented lattices, which in turn are a special case of complemented lattices (with extra structure).

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⊥.

Garrett Birkhoff and John von Neumann observed that the propositional calculus in quantum logic is "formally indistinguishable from the calculus of linear subspaces [of a Hilbert space] with respect to set products, linear sums and orthogonal complements" corresponding to the roles of and, or and not in Boolean lattices.

This remark has spurred interest in the closed subspaces of a Hilbert space, which form an orthomodular lattice.