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

$a\land b=0$ and $a\lor b=1$ .

Remark. Complements may not exist. If $L$ is a non-trivial chain, then no element (other than $0$ and $1$ ) has a complement. This also shows that if $a$ is a complement of a non-trivial 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.

$\xymatrix{&1\ar@{-}[ld]\ar@{-}[d]\ar@{-}[rd]&\\ a\ar@{-}[rd]&b\ar@{-}[d]&c\ar@{-}[ld]\\ &0&}$

Note that none of the non-trivial elements have unique complements. Any two non-trivial 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

$y_{2}=1\land y_{2}=(x\lor y_{1})\land y_{2}=(x\land y_{2})\lor(y_{1}\land y_{2% })=0\lor(y_{1}\land y_{2})=y_{1}\land y_{2}.$

Similarly, $y_{1}=y_{2}\land y_{1}$ . 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	2013-03-22 15:02:25
Last modified on	2013-03-22 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