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Homemodular lattice
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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 $x,y\in L$.
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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.