semimodular lattice

A lattice $L$ is semimodular¹¹Or upper semimodular, if one wants to stress thedistinction 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 semimodularif 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 $a\\wedge(c\\vee d)\eq(a\\wedge c)\\vee d$ .