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单词 ModularLattice
释义

modular lattice


A latticeMathworldPlanetmath L is said to be modularif x(yz)=(xy)zfor all x,y,zL such that xz.In fact it is sufficient to show thatx(yz)(xy)zfor all x,y,zL such that xz,as the reverse inequality holds in all lattices (see modular inequality).

There are a number of other equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath conditions for a lattice L to be modular:

  • (xy)(xz)=x(y(xz))for all x,y,zL.

  • (xy)(xz)=x(y(xz))for all x,y,zL.

  • For all x,y,zL,if x<z then either xy<zy or xy<zy.

The following are examples of modular lattices.

  • All distributive lattices (http://planetmath.org/DistributiveLattice).

  • The lattice of normal subgroupsMathworldPlanetmath of any group.

  • The lattice of submodules of any module (http://planetmath.org/Module).(See modular law.)

A finite lattice L is modularif and only if itis graded and its rank function ρ satisfiesρ(x)+ρ(y)=ρ(xy)+ρ(xy) for all x,yL.

Titlemodular lattice
Canonical nameModularLattice
Date of creation2013-03-22 12:27:26
Last modified on2013-03-22 12:27:26
Owneryark (2760)
Last modified byyark (2760)
Numerical id17
Authoryark (2760)
Entry typeDefinition
Classificationmsc 06C05
SynonymDedekind lattice
Related topicModularLaw
Related topicSemimodularLattice
Related topicNonmodularSublattice
Related topicModularInequality
Definesmodular
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更新时间:2025/5/4 20:30:39