Editing Complemented lattice (section)

==Orthocomplementation==
{{see also|De Morgan algebra}}
{{refimprove|section|date=August 2014}}
{{cleanup|reason=there are various competing definitions of "Orthocomplementation" in literature|date=August 2014}}
An '''orthocomplementation''' on a bounded lattice is a function that maps each element ''a'' to an "orthocomplement" ''a''<sup>⊥</sup> in such a way that the following axioms are satisfied:<ref>{{harvtxt|Stern|1999}}, p.&nbsp;11.</ref>
;Complement law: ''a''<sup>⊥</sup> ∨ ''a'' = 1 and ''a''<sup>⊥</sup> ∧ ''a'' = 0.
;Involution law: ''a''<sup>⊥⊥</sup> = ''a''.
;Order-reversing: if ''a'' ≤ ''b'' then ''b''<sup>⊥</sup> ≤ ''a''<sup>⊥</sup>.
An '''orthocomplemented lattice''' or '''ortholattice''' is a bounded lattice equipped with an orthocomplementation. The lattice of subspaces of an [[inner product space]], and the [[orthogonal complement]] operation, provides an example of an orthocomplemented lattice that is not, in general, distributive.<ref>[http://unapologetic.wordpress.com/2009/05/07/orthogonal-complements-and-the-lattice-of-subspaces/ The Unapologetic Mathematician: Orthogonal Complements and the Lattice of Subspaces].</ref>

<gallery Caption="Some complemented lattices">
Image:Smallest_nonmodular_lattice_1.svg|In the pentagon lattice ''N''<sub>5</sub>, the node on the right-hand side has two complements.
Image:Diamond lattice.svg|The diamond lattice ''M''<sub>3</sub> admits no orthocomplementation.
Image:Lattice M4.svg|The lattice ''M''<sub>4</sub> admits 3 orthocomplementations.
Image:Hexagon lattice.svg|The hexagon lattice admits a unique orthocomplementation, but it is not uniquely complemented.
</gallery>

[[Boolean algebra (structure)|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 set|closed]] [[Linear subspace|subspaces]] of a [[Separable space|separable]] [[Hilbert space]] represent quantum propositions and behave as an orthocomplemented lattice.

Orthocomplemented lattices, like Boolean algebras, satisfy [[de Morgan's laws]]:

* (''a'' ∨ ''b'')<sup>⊥</sup> = ''a''<sup>⊥</sup> ∧ ''b''<sup>⊥</sup>
* (''a'' ∧ ''b'')<sup>⊥</sup> = ''a''<sup>⊥</sup> ∨ ''b''<sup>⊥</sup>.