| Version 13 |
Version 12 |
| \PMlinkescapeword{modular} |
\PMlinkescapeword{modular} |
| \PMlinkescapeword{rank} |
\PMlinkescapeword{rank} |
| \PMlinkescapeword{satisfies} |
\PMlinkescapeword{satisfies} |
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| A lattice $L$ is said to be \emph{modular} |
A lattice $L$ is said to be \emph{modular} if $x \lor (y \land z) = (x \lor y) \land z$ for all $x,y,z\in L$ such that $x \leq z$. |
| if $x \lor (y \land z) = (x \lor y) \land z$ |
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| for all $x,y,z\in L$ such that $x \leq z$. |
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| In fact it is only necessary to show that |
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| $x \lor (y \land z) \ge (x \lor y) \land z$ |
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| for all $x,y,z\in L$ such that $x \leq z$, |
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| as the reverse inequality holds in all lattices (see modular inequality). |
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| There are a number of other equivalent conditions for a lattice $L$ to be modular: |
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| \begin{itemize} |
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| \item $(x\meet y)\join(x\meet z)=x\meet(y\join(x\meet z))$ |
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| for all $x,y,z\in L$. |
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| \item $(x\join y)\meet(x\join z)=x\join(y\meet(x\join z))$ |
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| for all $x,y,z\in L$. |
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| \item For all $x,y,z\in L$, |
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| if $x<z$ then either $x\meet y<z\meet y$ or $x\join y<z\join y$. |
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| \end{itemize} |
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| The following are examples of modular lattices. |
The following are examples of modular lattices. |
| \begin{itemize} |
\begin{itemize} |
| \item All \PMlinkname{distributive lattices}{DistributiveLattice}. |
\item All \PMlinkname{distributive lattices}{DistributiveLattice}. |
| \item The lattice of normal subgroups of any group. |
\item The lattice of normal subgroups of any group. |
| \item The lattice of submodules of any \PMlinkname{module}{Module}. |
\item The lattice of submodules of any \PMlinkname{module}{Module}. (See modular law.) |
| (See modular law.) |
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| \end{itemize} |
\end{itemize} |
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| A finite lattice $L$ is modular |
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 $x,y\in L$. |
| if and only if it |
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| is graded and its rank function $\rho$ satisfies |
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| $\rho(x)+\rho(y)=\rho(x\land y)+\rho(x\lor y)$ for all $x,y\in L$. |
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