In the branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
known as
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, 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)
matroid
In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most significant being in terms ...
s. An atomistic semimodular bounded lattice of finite length is called a
geometric lattice
In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric lattices and matroid lattices, ...
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
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
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.
[For instance, Fofanova (2001).]
A semimodular lattice is one kind of
algebraic lattice.
Birkhoff's condition
A lattice is sometimes called weakly semimodular if it satisfies the following condition due to
Garrett Birkhoff
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory.
The mathematician George Birkhoff (1884–1944) was his father.
Life
The son of the mathematician Ge ...
:
;Birkhoff's condition: If ''a'' ∧ ''b'' <: ''a'' and ''a'' ∧ ''b'' <: ''b'',
:then ''a'' <: ''a'' ∨ ''b'' and ''b'' <: ''a'' ∨ ''b''.
Every semimodular lattice is weakly semimodular. The converse is true for lattices of finite length, and more generally for upper continuous (meets distribute over joins of chains)
relatively atomic lattices.
Mac Lane's condition
The following two conditions are equivalent to each other for all lattices. They were found by
Saunders Mac Lane
Saunders Mac Lane (August 4, 1909 – April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg.
Early life and education
Mac Lane was born in Norwich, Connecticut, near w ...
, who was looking for a condition that is equivalent to semimodularity for finite lattices, but does not involve the covering relation.
;Mac Lane's condition 1: For any ''a, b, c'' such that ''b'' ∧ ''c'' < ''a'' < ''c'' < ''b'' ∨ ''a'',
:there is an element ''d'' such that ''b'' ∧ ''c'' < ''d'' ≤ ''b'' and ''a'' = (''a'' ∨ ''d'') ∧ ''c''.
;Mac Lane's condition 2: For any ''a, b, c'' such that ''b'' ∧ ''c'' < ''a'' < ''c'' < ''b'' ∨ ''c'',
:there is an element ''d'' such that ''b'' ∧ ''c'' < ''d'' ≤ ''b'' and ''a'' = (''a'' ∨ ''d'') ∧ ''c''.
Every lattice satisfying Mac Lane's condition is semimodular. The converse is true for lattices of finite length, and more generally for
relatively atomic lattices. Moreover, every upper continuous lattice satisfying Mac Lane's condition is M-symmetric.
Notes
References
* . (The article is about M-symmetric lattices.)
* .
External links
*
*
See also
*
Antimatroid
{{DEFAULTSORT:Semimodular Lattice
Lattice theory