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  20130322 12:27:26 
Last modified on  20130322 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 