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单词 ComplementedLattice
释义

complemented lattice


Let L be a bounded latticeMathworldPlanetmath (with 0 and 1), and aL. A complementPlanetmathPlanetmath of a is an element bL such that

ab=0 and ab=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 antichainMathworldPlanetmath.

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.

  • 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 latticeMathworldPlanetmath 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.

  • If a complemented lattice L is a distributive latticeMathworldPlanetmath, then L is uniquely complemented (in fact, a Boolean lattice). For if y1 and y2 are two complements of x, then

    y2=1y2=(xy1)y2=(xy2)(y1y2)=0(y1y2)=y1y2.

    Similarly, y1=y2y1. So y2=y1.

  • In the category of complemented lattices, a morphism between two objects is a {0,1}-lattice homomorphismMathworldPlanetmath; that is, a lattice homomorphism which preserves 0 and 1.

Titlecomplemented lattice
Canonical nameComplementedLattice
Date of creation2013-03-22 15:02:25
Last modified on2013-03-22 15:02:25
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id26
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 06C15
Classificationmsc 06B05
Synonymperspective elements
Synonymcomplemented
Related topicPerspectivity
Related topicOrthocomplementedLattice
Related topicPseudocomplementedLattice
Related topicDifferenceOfLatticeElements
Related topicPseudocomplement
Definesrelated elements in lattice
Definescomplement
Definescomplemented element
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更新时间:2025/5/4 1:00:49