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'modular lattice'
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| Title of object: |
modular lattice |
| Canonical Name: |
ModularLattice |
| Type: |
Definition |
| Created on: |
2002-02-24 15:11:23 |
| Modified on: |
2004-02-24 04:46:55 |
| Classification: |
msc:06C05 |
Preamble:
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Content:
\PMlinkescapeword{rank}
\PMlinkescapeword{satisfies}
A lattice $L$ is said to be {\em modular} if $x \lor (y \land z) = (x \lor y) \land z$ for all $x,y,z\in L$ such that $x \leq z$.
The following are examples of modular lattices.
\begin{itemize}
\item All \PMlinkname{distributive lattices}{DistributiveLattice}.
\item The lattice of normal subgroups of any group.
\item The lattice of submodules of any \PMlinkname{module}{Module}. (See modular law.)
\end{itemize}
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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