Semimodular lattice

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.
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 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.

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.

A semimodular lattice is one kind of algebraic lattice.

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