Editing Complemented lattice (section)

{{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]].