# semimodular lattice

A lattice $L$ is semimodular 11Or 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 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 modular (http://planetmath.org/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 $d\leq a$ but $a\wedge(c\vee d)\neq(a\wedge c)\vee d$.

Title semimodular lattice SemimodularLattice 2013-03-22 15:26:20 2013-03-22 15:26:20 mps (409) mps (409) 9 mps (409) Definition msc 06C10 upper semimodular lattice lower semimodular lattice ModularLattice IncidenceGeometry