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 bL,


where denotes the covering relation in L. Dually, a lattice L is said to be lower semimodular if for any a and bL,


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

The smallest lattice which is semimodular but not modular is


since da but a(cd)(ac)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