semimodular latticeSemimodularLattice2005-08-02 08:47:542007-04-12 08:49:25Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA lattice $L$ is {\em semimodular}
\footnote{Or {\em upper semimodular}, if one wants to stress the
distinction with lower semimodular lattices.}
if for any $a$ and $b\in L$,
\[
a \wedge b \prec a
\quad\text{implies}\quad
b \prec a \vee b,
\]
where $\prec$ denotes the covering relation in $L$.
Dually, a lattice $L$ is said to be \emph{lower semimodular}
if for any $a$ and $b\in L$,
\[
b \prec a \vee b
\quad\text{implies}\quad
a \wedge b \prec a.
\]
A chain finite lattice is \PMlinkname{modular}{ModularLattice}
if and only if it is both semimodular and lower semimodular.
The smallest lattice which is semimodular but not modular is
\[\xymatrix{
& 1 \ar@{-}[ld] \ar@{-}[d] \ar@{-}[rd] & \\
a \ar@{-}[d] & b \ar@{-}[ld] \ar@{-}[rd] & c \ar@{-}[d] \\
d \ar@{-}[rd] & & e \ar@{-}[ld] \\
& 0 &
}\]
since $d \le a$ but $a \wedge (c \vee d) \neq (a \wedge c) \vee d$.