modular latticeModularLattice2002-02-24 15:11:232007-04-20 12:04:22Definitionmodular\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\def\meet{\land}
\def\join{\lor}
\PMlinkescapeword{modular}
\PMlinkescapeword{rank}
\PMlinkescapeword{satisfies}
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 $x \leq z$.
In fact it is sufficient to show that
$x \lor (y \land z) \ge (x \lor y) \land z$
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:
\begin{itemize}
\item $(x\meet y)\join(x\meet z)=x\meet(y\join(x\meet z))$
for all $x,y,z\in L$.
\item $(x\join y)\meet(x\join z)=x\join(y\meet(x\join z))$
for all $x,y,z\in L$.
\item For all $x,y,z\in L$,
if $x<z$ then either $x\meet y<z\meet y$ or $x\join y<z\join y$.
\end{itemize}
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 $x,y\in L$.