Complemented Lattice - Orthomodular Lattices

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. Garrett Birkhoff and John von Neumann observed that the propositional calculus in quantum logic is "formally indistinguishable from the calculus of linear subspaces 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.