Semimodular lattice

A finite lattice is modular if and only if it is both upper and lower semimodular.

The converse is true for lattices of finite length, and more generally for upper continuous (meets distribute over joins of chains) relatively atomic lattices.

They were found by Saunders Mac Lane, who was looking for a condition that is equivalent to semimodularity for finite lattices, but does not involve the covering relation.

Every lattice satisfying Mac Lane's condition is semimodular.

Moreover, every upper continuous lattice satisfying Mac Lane's condition is M-symmetric.

The centred hexagon lattice S 7 , also known as D 2 , is semimodular but not modular.