Editing Complemented lattice (section)

==Orthomodular lattices==

A lattice is called [[modular lattice|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 [[Distributive lattice|distributivity]]; e.g. the above-shown lattice ''M''<sub>3</sub> 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''<sup>⊥</sup>. An '''orthomodular lattice''' is therefore defined as an orthocomplemented lattice such that for any two elements the implication
::if ''a'' ≤ ''c'', then ''a'' ∨ (''a''<sup>⊥</sup> ∧ ''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]] [[mathematical formulation of quantum mechanics|formulation]] of [[quantum mechanics]]. [[Garrett Birkhoff]] and [[John von Neumann]] observed that the [[propositional logic|propositional]] [[logical calculus|calculus]] in quantum logic is "formally indistinguishable from the calculus of linear subspaces [of a Hilbert space] with respect to [[intersection|set products]], [[Linear_subspace#Sum|linear sum]]s 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.<ref name="PadmanabhanRudeanu2008">{{cite book|author1=Ranganathan Padmanabhan|author2=Sergiu Rudeanu|title=Axioms for lattices and boolean algebras|url=https://books.google.com/books?id=JlXSlpmlSv4C&pg=PA128|year=2008|publisher=World Scientific|isbn=978-981-283-454-6|page=128}}</ref>