complemented lattice
Let $L$ be a bounded lattice^{} (with $0$ and $1$), and $a\in L$. A complement^{} of $a$ is an element $b\in L$ such that
$a\wedge b=0$ and $a\vee b=1$.
Remark. Complements may not exist. If $L$ is a nontrivial chain, then no element (other than $0$ and $1$) has a complement. This also shows that if $a$ is a complement of a nontrivial element $b$, then $a$ and $b$ form an antichain^{}.
An element in a bounded lattice is complemented if it has a complement. A complemented lattice is a bounded lattice in which every element is complemented.
Remarks.

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In a complemented lattice, there may be more than one complement corresponding to each element. Two elements are said to be related, or perspective if they have a common complement. For example, the following lattice^{} is complemented.
$$\text{xymatrix}\mathrm{\&}1\text{ar}\mathrm{@}[ld]\text{ar}\mathrm{@}[d]\text{ar}\mathrm{@}[rd]\mathrm{\&}a\text{ar}\mathrm{@}[rd]\mathrm{\&}b\text{ar}\mathrm{@}[d]\mathrm{\&}c\text{ar}\mathrm{@}[ld]\mathrm{\&}0\mathrm{\&}$$ Note that none of the nontrivial elements have unique complements. Any two nontrivial elements are related via the third.

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If a complemented lattice $L$ is a distributive lattice^{}, then $L$ is uniquely complemented (in fact, a Boolean lattice). For if ${y}_{1}$ and ${y}_{2}$ are two complements of $x$, then
$${y}_{2}=1\wedge {y}_{2}=(x\vee {y}_{1})\wedge {y}_{2}=(x\wedge {y}_{2})\vee ({y}_{1}\wedge {y}_{2})=0\vee ({y}_{1}\wedge {y}_{2})={y}_{1}\wedge {y}_{2}.$$ Similarly, ${y}_{1}={y}_{2}\wedge {y}_{1}$. So ${y}_{2}={y}_{1}$.

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In the category of complemented lattices, a morphism between two objects is a $\{0,1\}$lattice homomorphism^{}; that is, a lattice homomorphism which preserves $0$ and $1$.
Title  complemented lattice 
Canonical name  ComplementedLattice 
Date of creation  20130322 15:02:25 
Last modified on  20130322 15:02:25 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  26 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06C15 
Classification  msc 06B05 
Synonym  perspective elements 
Synonym  complemented 
Related topic  Perspectivity 
Related topic  OrthocomplementedLattice 
Related topic  PseudocomplementedLattice 
Related topic  DifferenceOfLatticeElements 
Related topic  Pseudocomplement 
Defines  related elements in lattice 
Defines  complement 
Defines  complemented element 