relative complement
A complement of an element in a lattice
is only defined when the lattice in question is bounded (http://planetmath.org/BoundedLattice). In general, a lattice is not bounded and there are no complements to speak of. Nevertheless, if the sublattice of a lattice is bounded, we can speak of complements of an element relative to that sublattice.
Let L be a lattice, a an element of L, and I=[b,c] an interval (http://planetmath.org/LatticeInterval) in L. An element d∈L is said to be a complement of a relative to I if
a∨d=c and a∧d=b. |
It is easy to see that a≤c and b≤a, so a∈I. Similarly, d∈I.
An element a∈L is said to be relatively complemented if for every interval I in L with a∈I, it has a complement relative to I. The lattice L itself is called a relatively complemented lattice if every element of L is relatively complemented. Equivalently, L is relatively complemented iff each of its interval is a complemented lattice.
Remarks.
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A relatively complemented lattice is complemented if it is bounded. Conversely, a complemented lattice is relatively complemented if it is modular (http://planetmath.org/ModularLattice).
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The notion of a relative complement of an element in a lattice has nothing to do with that found in set theory
: let U be a set and A,B subsets of U, the relative complement of A in B is the set theoretic difference
B-A. While the relative difference is necessarily a subset of B, A does not have to be a subset of B.
Title | relative complement |
---|---|
Canonical name | RelativeComplement |
Date of creation | 2013-03-22 15:51:45 |
Last modified on | 2013-03-22 15:51:45 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 06C15 |
Related topic | RelativePseudocomplement |
Related topic | BrouwerianLattice |
Defines | relatively complemented lattice |
Defines | relatively complemented |