Semimodular lattice

This article is about generalizations of modularity in terms of the atomic covering relation. For M-symmetry, the generalization of modularity in terms of modular pairs, see modular lattice.
The centred hexagon lattice S7, also known as D2, is semimodular but not modular.

In the branch of mathematics known as order theory, a semimodular lattice, is a lattice that satisfies the following condition:

Semimodular law
a ∧ b  <:  a   implies   b  <:  a ∨ b.

The notation a <: b means that b covers a, i.e. a < b and there is no element c such that a < c < b.

An atomistic semimodular bounded lattice is called a matroid lattice because such lattices are equivalent to (simple) matroids. An atomistic semimodular bounded lattice of finite length is called a geometric lattice and corresponds to a matroid of finite rank.[1]

Semimodular lattices are also known as upper semimodular lattices; the dual notion is that of a lower semimodular lattice. A finite lattice is modular if and only if it is both upper and lower semimodular.

A finite lattice, or more generally a lattice satisfying the ascending chain condition or the descending chain condition, is semimodular if and only if it is M-symmetric. Some authors refer to M-symmetric lattices as semimodular lattices.[2]

A semimodular lattice is one kind of algebraic lattice.

  1. ^ These definitions follow Stern (1999). Some authors use the term geometric lattice for the more general matroid lattices. Most authors only deal with the finite case, in which both definitions are equivalent to semimodular and atomistic.
  2. ^ For instance, Fofanova (2001).