of an inner product space, and the orthogonal complement operation, provides an example of an orthocomplemented lattice that is not, in general, distributive. Boolean algebras are a special case of orthocomplemented lattices, which in turn are a special case of complemented lattices (with extra structure). The ortholattices 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⊥. 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; e.g. the above-shown lattice M3 is modular, but not distributive. 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. 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. See also Pseudocomplemented lattice Notes References External links Lattice theory