modular lattice

A latticeMathworldPlanetmath L is said to be modular if x(yz)=(xy)z for all x,y,zL such that xz. In fact it is sufficient to show that x(yz)(xy)z for 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 (

  • The lattice of normal subgroupsMathworldPlanetmath of any group.

  • The lattice of submodules of any module ( (See modular law.)

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

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