complemented lattice
Let L be a bounded lattice (with 0 and 1), and a∈L. A complement
of a is an element b∈L such that
a∧b=0 and a∨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 y1 and y2 are two complements of x, then
y2=1∧y2=(x∨y1)∧y2=(x∧y2)∨(y1∧y2)=0∨(y1∧y2)=y1∧y2. Similarly, y1=y2∧y1. So y2=y1.
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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 |