modular lattice

A lattice $L$ is said to be modular if $x\vee (y\wedge z)=(x\vee y)\wedge z$ for all $x,y,z\in L$ such that $x\le z$. In fact it is sufficient to show that $x\vee (y\wedge z)\ge (x\vee y)\wedge z$ for all $x,y,z\in L$ such that $x\le z$, as the reverse inequality holds in all lattices (see modular inequality).

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

• $(x\wedge y)\vee (x\wedge z)=x\wedge (y\vee (x\wedge z))$ for all $x,y,z\in L$.

• $(x\vee y)\wedge (x\vee z)=x\vee (y\wedge (x\vee z))$ for all $x,y,z\in L$.

• For all $x,y,z\in L$, if then either or .

The following are examples of modular lattices.

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

• The lattice of normal subgroups of any group.

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

A finite lattice $L$ is modular if and only if it is graded and its rank function $\rho$ satisfies $\rho (x)+\rho (y)=\rho (x\wedge y)+\rho (x\vee y)$ for all $x,y\in L$.

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