You are here
Homemodular lattice
Primary tabs
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 $x<z$ then either $x\land y<z\land y$ or $x\lor y<z\lor y$.
The following are examples of modular lattices.

The lattice of normal subgroups 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 $\rho$ satisfies $\rho(x)+\rho(y)=\rho(x\land y)+\rho(x\lor y)$ for all $x,y\in L$.
Mathematics Subject Classification
06C05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Comments
modular inequality
Maybe the article should mention that in order to show that a lattice is modular it is really only necessary to show that x<=z implies xâˆ¨(yâˆ§z)>=(xâˆ¨y)âˆ§z since x<=z implies xâˆ¨(yâˆ§z)<=(xâˆ¨y)âˆ§z for any lattice.
Re: modular inequality
If you would like the entry to be changed, please post a correction (preferably without strange characters).
Re: modular inequality
Feel free to add "modular inequality" to the encyclopedia, with "modular lattice" as the parent. Also, prove this inequality in your entry, if possible.
Re: modular inequality
should "modular inequality" not be attached to "lattice", since it applies to any lattice and not just modular lattices?
Re: modular inequality
It is true that modular inequality applies to any lattice. But so do other lattice inequalities, like the distributive inequalities, which are covered under a separate entry whose parent is "distributive lattice".
I would recommend either putting it in a separate entry (with modular lattice as a parent), or file a correction to the "modular lattice" entry so it is mentioned under there.