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.