Semimodular lattice

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

semimodular implication: a ∧ b :> a implies b :> a ∨ b.

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

An algebraic atomistic semimodular bounded lattice is called a matroid lattice because such lattices are equivalent to 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]

References

^ These definitions follow Stern (1999). Some authors use the term geometric lattice for the more general matroid lattices.
^ For instance Fofanova (2001).

External links

"Geometric lattice". PlanetMath.. (The article is about matroid lattices.)
"Semimodular lattice". PlanetMath..

References

Fofanova, T. S. (2001) [1994], "Semi-modular lattice", Encyclopedia of Mathematics, EMS Press. (The article is about M-symmetric lattices.)
Stern, Manfred (1999), Semimodular lattices, Cambridge University Press, ISBN 978-0-521-46105-4.

[1] These definitions follow Stern (1999). Some authors use the term geometric lattice for the more general matroid lattices.

[2] For instance Fofanova (2001).

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