Editing Complemented lattice

{{Short description|Bound lattice in which every element has a complement}}
[[File:Fano plane Hasse diagram.svg|thumb|[[Hasse diagram]] of a complemented lattice. A point {{mvar|p}} and a line {{mvar|l}} of the [[Fano plane]] are complements if and only if {{mvar|p}} does not lie on {{mvar|l}}.]]
In the [[mathematics|mathematical]] discipline of [[order theory]], a '''complemented lattice''' is a bounded [[lattice (order)|lattice]] (with [[least element]] 0 and [[greatest element]] 1), in which every element ''a'' has a '''complement''', i.e. an element ''b'' satisfying ''a''&nbsp;∨&nbsp;''b''&nbsp;=&nbsp;1 and ''a''&nbsp;∧&nbsp;''b''&nbsp;=&nbsp;0.
Complements need not be unique.

A '''relatively complemented lattice''' is a lattice such that every [[Interval (partial order)|interval]] [''c'',&nbsp;''d''], viewed as a bounded lattice in its own right, is a complemented lattice.

An '''orthocomplementation''' on a complemented lattice is an [[involution (mathematics)|involution]] that is [[order-reversing]] and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the [[modular lattice|modular law]] is called an '''orthomodular lattice'''.

In bounded [[distributive lattice]]s, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a [[Boolean algebra (structure)|Boolean algebra]].

==Definition and basic properties==
A '''complemented lattice''' is a bounded lattice (with [[least element]] 0 and [[greatest element]] 1), in which every element ''a'' has a '''complement''', i.e. an element ''b'' such that
::''a'' ∨ ''b'' = 1 &nbsp;&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;&nbsp;&nbsp;''a'' ∧ ''b'' = 0.

In general an element may have more than one complement. However, in a (bounded) [[distributive lattice]] every element will have at most one complement.<ref>Grätzer (1971), Lemma I.6.1, p. 47. Rutherford (1965), Theorem 9.3 p. 25.</ref> A lattice in which every element has exactly one complement is called a '''uniquely complemented lattice'''<ref>{{citation|title=Semimodular Lattices: Theory and Applications|series=Encyclopedia of Mathematics and its Applications|first=Manfred|last=Stern|publisher=Cambridge University Press|year=1999|isbn=9780521461054|page=29|url=https://books.google.com/books?id=VVYd2sC19ogC&pg=PA29}}.</ref>

A lattice with the property that every interval (viewed as a sublattice) is complemented is called a '''relatively complemented lattice'''. In other words, a relatively complemented lattice is characterized by the property that for every element ''a'' in an interval [''c'', ''d''] there is an element ''b'' such that
::''a'' ∨ ''b'' = ''d'' &nbsp;&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;&nbsp;&nbsp;''a'' ∧ ''b'' = ''c''.
Such an element ''b'' is called a complement of ''a'' relative to the interval. 

A distributive lattice is complemented if and only if it is bounded and relatively complemented.<ref>Grätzer (1971), Lemma I.6.2, p. 48. This result holds more generally for modular lattices, see Exercise 4, p. 50.</ref><ref>Birkhoff (1961), Corollary IX.1, p. 134</ref> The lattice of [[vector subspace|subspace]]s of a [[vector space]] provide an example of a complemented lattice that is not, in general, distributive.

==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>.

==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>

==See also==
* [[Pseudocomplemented lattice]]

==Notes==
{{Reflist}}

==References==
* {{Cite book | last1=Birkhoff | first1=Garrett | title=Lattice Theory | publisher=American Mathematical Society | year=1961 }}
* {{Cite book | last1=Grätzer | first1=George |authorlink = George Grätzer | title=Lattice Theory: First Concepts and Distributive Lattices | publisher=W. H. Freeman and Company | isbn=978-0-7167-0442-3 | year=1971 }}
* {{Cite book | last1=Grätzer | first1=George | title=General Lattice Theory | publisher=Birkhäuser | location=Basel, Switzerland | isbn=978-0-12-295750-5 | year=1978 }}
* {{cite book | first = Daniel Edwin | last = Rutherford | year = 1965 | title = Introduction to Lattice Theory | publisher = Oliver and Boyd }}

==External links==

{| style="float:right"
| {{Algebraic structures |Lattice}}
|}
* {{planetmath reference|urlname=ComplementedLattice|title=Complemented lattice}}
* {{planetmath reference|urlname=RelativeComplement|title=Relative complement}}
* {{planetmath reference|urlname=UniquelyComplementedLattice|title=Uniquely complemented lattice}}
* {{planetmath reference|urlname=OrthocomplementedLattice|title=Orthocomplemented lattice}}

{{Order theory}}

[[Category:Lattice theory]]