Editing Complemented lattice (section)

==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.