Orthomodular Lattice

An orthomodular lattice is a lattice equipped with an orthocomplementation operation — a map a ↦ a⊥ satisfying a ∨ a⊥ = 1, a ∧ a⊥ = 0, and (a⊥)⊥ = a — that satisfies the orthomodular law: if a ≤ b, then b = a ∨ (b ∧ a⊥). This law is strictly weaker than the distributive law that defines Boolean algebras, and strictly stronger than the modular law that defines modular lattices. It occupies a precise structural niche: orthomodular lattices are exactly the lattices that can arise as the lattice of closed subspaces of a Hilbert space, which makes them the native logical structure of quantum logic. The orthomodular law is not an arbitrary weakening of distributivity. It is the minimal structural condition needed to support probability measures that behave like quantum states. Any lattice that violates orthomodularity cannot support a sensible quantum probability theory.