In the mathematics of
matroid
In combinatorics, a branch of mathematics, 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 axiomatically, the most significant being ...
s and
lattices, a geometric lattice is a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
atomistic 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'' & ...
, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of
flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way.
Definition
A
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
is a
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
in which any two elements
and
have both a least upper bound, called the join or
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
, denoted by
, and a greatest lower bound, called the meet or
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
, denoted by
.
: The following definitions apply to posets in general, not just lattices, except where otherwise stated.
* For a
minimal element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defin ...
, there is no element
such that
.
* An element
covers another element
(written as
or
) if
and there is no element
distinct from both
and
so that
.
* A cover of a minimal element is called an
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, a ...
.
* A lattice is
atomistic if every element is the supremum of some set of atoms.
* A poset is
graded when it can be given a rank function
mapping its elements to integers, such that
whenever
, and also
whenever
.
: When a graded poset has a bottom element, one may assume, without loss of generality, that its rank is zero. In this case, the atoms are the elements with rank one.
* A graded lattice is
semimodular if, for every
and
, its rank function obeys the identity
::
* A matroid lattice is a lattice that is both atomistic and semimodular. A geometric lattice is a ''finite'' matroid lattice.
[, p. 51.]
: Many authors consider only finite matroid lattices, and use the terms "geometric lattice" and "matroid lattice" interchangeably for both.
Lattices vs. matroids
The geometric lattices are equivalent to (finite, simple) matroids, and the matroid lattices are equivalent to simple matroids without the assumption of finiteness (under an appropriate definition of infinite matroids; there are several such definitions). The correspondence is that the elements of the matroid are the atoms of the lattice and an element ''x'' of the lattice corresponds to the flat of the matroid that consists of those elements of the matroid that are atoms
Like a geometric lattice, a matroid is endowed with a
rank function, but that function maps a set of matroid elements to a number rather than taking a lattice element as its argument. The rank function of a matroid must be monotonic (adding an element to a set can never decrease its rank) and it must be
submodular
In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an ...
, meaning that it obeys an inequality similar to the one for semimodular ranked lattices:
:
for sets ''X'' and ''Y'' of matroid elements.
The
maximal sets of a given rank are called flats. The intersection of two flats is again a flat, defining a greatest lower bound operation on pairs of flats; one can also define a least upper bound of a pair of flats to be the (unique) maximal superset of their union that has the same rank as their union. In this way, the flats of a matroid form a matroid lattice, or (if the matroid is finite) a geometric lattice.
Conversely, if
is a matroid lattice, one may define a rank function on sets of its atoms, by defining the rank of a set of atoms to be the lattice rank of the greatest lower bound of the set. This rank function is necessarily monotonic and submodular, so it defines a matroid. This matroid is necessarily simple, meaning that every two-element set has rank two.
These two constructions, of a simple matroid from a lattice and of a lattice from a matroid, are inverse to each other: starting from a geometric lattice or a simple matroid, and performing both constructions one after the other, gives a lattice or matroid that is isomorphic to the original one.
Duality
There are two different natural notions of duality for a geometric lattice
: the dual matroid, which has as its basis sets the
complements of the bases of the matroid corresponding to
, and the
dual lattice
In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a lattice L is the reciprocal of the geometry of L , a perspective which underlie ...
, the lattice that has the same elements as
in the reverse order. They are not the same, and indeed the dual lattice is generally not itself a geometric lattice: the property of being atomistic is not preserved by order-reversal. defines the
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
of a geometric lattice
(or of the matroid defined from it) to be a minimal geometric lattice into which the dual lattice of
is
order-embedded. Some matroids do not have adjoints; an example is the
Vámos matroid
In mathematics, the Vámos matroid or Vámos cube is a matroid over a set of eight elements that cannot be represented as a matrix over any field. It is named after English mathematician Peter Vámos, who first described it in an unpublished ma ...
.
Additional properties
Every interval of a geometric lattice (the subset of the lattice between given lower and upper bound elements) is itself geometric; taking an interval of a geometric lattice corresponds to forming a
minor of the associated matroid. Geometric lattices are
complemented, and because of the interval property they are also relatively complemented.
Every finite lattice is a sublattice of a geometric lattice.
[, p. 58; Welsh credits this result to Robert P. Dilworth, who proved it in 1941–1942, but does not give a specific citation for its original proof.]
References
External links
*
* {{OEIS el, A281574, Number of unlabeled geometric lattices with ''n'' elements
Lattice theory
Matroid theory