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
Canonical name	SemimodularLattice
Date of creation	2013-03-22 15:26:20
Last modified on	2013-03-22 15:26:20
Owner	mps (409)
Last modified by	mps (409)
Numerical id	9
Author	mps (409)
Entry type	Definition
Classification	msc 06C10
Synonym	upper semimodular lattice
Synonym	lower semimodular lattice
Related topic	ModularLattice
Related topic	IncidenceGeometry