# modular lattice

A lattice $L$ is said to be modular if $x\lor(y\land z)=(x\lor y)\land z$ for all $x,y,z\in L$ such that $x\leq z$. In fact it is sufficient to show that $x\lor(y\land z)\geq(x\lor y)\land z$ for all $x,y,z\in L$ such that $x\leq z$, 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:

• $(x\land y)\lor(x\land z)=x\land(y\lor(x\land z))$ for all $x,y,z\in L$.

• $(x\lor y)\land(x\lor z)=x\lor(y\land(x\lor z))$ for all $x,y,z\in L$.

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

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 $\rho(x)+\rho(y)=\rho(x\land y)+\rho(x\lor y)$ 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