modular lattice
A lattice is said to be modularif for all such that .In fact it is sufficient to show thatfor all such that ,as the reverse inequality holds in all lattices (see modular inequality).
There are a number of other equivalent conditions for a lattice to be modular:
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for all .
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for all .
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For all ,if then either or .
The following are examples of modular lattices.
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All distributive lattices (http://planetmath.org/DistributiveLattice).
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The lattice of normal subgroups
of any group.
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The lattice of submodules of any module (http://planetmath.org/Module).(See modular law.)
A finite lattice is modularif and only if itis graded and its rank function satisfies for all .
Title | modular lattice |
Canonical name | ModularLattice |
Date of creation | 2013-03-22 12:27:26 |
Last modified on | 2013-03-22 12:27:26 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 17 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 06C05 |
Synonym | Dedekind lattice |
Related topic | ModularLaw |
Related topic | SemimodularLattice |
Related topic | NonmodularSublattice |
Related topic | ModularInequality |
Defines | modular |