Formulation of matroids using closure operators
Pregeometry, and in full combinatorial pregeometry, are essentially synonyms for "
matroid". They were introduced by
Gian-Carlo Rota with the intention of providing a less "ineffably cacophonous" alternative term. Also, the term combinatorial geometry, sometimes abbreviated to geometry, was intended to replace "simple matroid". These terms are now infrequently used in the study of matroids.
It turns out that many fundamental concepts of
linear algebra – closure, independence, subspace, basis, dimension – are available in the general framework of pregeometries.
In the branch of
mathematical logic called
model theory, infinite finitary matroids, there called "pregeometries" (and "geometries" if they are simple matroids), are used in the discussion of independence phenomena. The study of how pregeometries, geometries, and abstract
closure operators influence the structure of
first-order models is called
geometric stability theory.
Motivation
If
is a
vector space over some field and
, we define
to be the set of all
linear combinations of vectors from
, also known as the
span of
. Then we have
and
and
. The
Steinitz exchange lemma is equivalent to the statement: if
, then
The linear algebra concepts of independent set, generating set, basis and dimension can all be expressed using the
-operator alone. A pregeometry is an abstraction of this situation: we start with an arbitrary set
and an arbitrary operator
which assigns to each subset
of
a subset
of
, satisfying the properties above. Then we can define the "linear algebra" concepts also in this more general setting.
This generalized notion of dimension is very useful in model theory, where in certain situation one can argue as follows: two models with the same cardinality must have the same dimension and two models with the same dimension must be isomorphic.
Definitions
Pregeometries and geometries
A combinatorial pregeometry (also known as a finitary matroid) is a pair
, where
is a set and
(called the closure map) satisfies the following axioms. For all
and
:
is monotone increasing and dominates
(i.e.
implies
) and is idempotent (i.e.
)
- Finite character: For each
there is some finite
with
.
- Exchange principle: If
, then
(and hence by monotonicity and idempotence in fact
).
Sets of the form
for some
are called closed. It is then clear that finite intersections of closed sets are closed and that
is the smallest closed set containing
.
A geometry is a pregeometry in which the closure of singletons are singletons and the closure of the empty set is the empty set.
Independence, bases and dimension
Given sets
,
is independent over
if
for any
. We say that
is independent if it is independent over the empty set.
A set
is a basis for
over
if it is independent over
and
.
A basis is the same as a maximal independent subset, and using
Zorn's lemma one can show that every set has a basis. Since a pregeometry satisfies the
Steinitz exchange property all bases are of the same cardinality, hence we may define the dimension of
over
, written as
, as the cardinality of any basis of
over
. Again, the dimension
of
is defined to be the dimesion over the empty set.
The sets
are independent over
if
whenever
is a finite subset of
. Note that this relation is symmetric.
Automorphisms and homogeneous pregeometries
An automorphism of a pregeometry
is a bijection
such that
for any
.
A pregeometry
is said to be homogeneous if for any closed
and any two elements
there is an automorphism of
which maps
to
and fixes
pointwise.
The associated geometry and localizations
Given a pregeometry
its associated geometry (sometimes referred in the literature as the canonical geometry) is the geometry
where
, and
- For any
, ![{\displaystyle {\text{cl}}'(\{{\text{cl}}(a)\mid a\in X\})=\{{\text{cl}}(b)\mid b\in {\text{cl}}(X)\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e51612c4e515e6e0bddd9ddc70b8023be88ecd3)
Its easy to see that the associated geometry of a homogeneous pregeometry is homogeneous.
Given
the localization of
is the pregeometry
where
.
Types of pregeometries
The pregeometry
is said to be:
- trivial (or degenerate) if
for all non-empty
.
- modular if any two closed finite dimensional sets
satisfy the equation
(or equivalently that
is independent of
over
).
- locally modular if it has a localization at a singleton which is modular.
- (locally) projective if it is non-trivial and (locally) modular.
- locally finite if closures of finite sets are finite.
Triviality, modularity and local modularity pass to the associated geometry and are preserved under localization.
If
is a locally modular homogeneous pregeometry and
then the localization of
in
is modular.
The geometry
is modular if and only if whenever
,
,
and
then
.
Examples
The trivial example
If
is any set we may define
for all
. This pregeometry is a trivial, homogeneous, locally finite geometry.
Vector spaces and projective spaces
Let
be a
field (a division ring actually suffices) and let
be a vector space over
. Then
is a pregeometry where closures of sets are defined to be their
span. The closed sets are the linear subspaces of
and the notion of dimension from linear algebra coincides with the pregeometry dimension.
This pregeometry is homogeneous and modular. Vector spaces are considered to be the prototypical example of modularity.
is locally finite if and only if
is finite.
is not a geometry, as the closure of any nontrivial vector is a subspace of size at least
.
The associated geometry of a
-dimensional vector space over
is the
-dimensional
projective space over
. It is easy to see that this pregeometry is a projective geometry.
Affine spaces
Let
be a
-dimensional
affine space over a field
. Given a set define its closure to be its
affine hull (i.e. the smallest affine subspace containing it).
This forms a homogeneous
-dimensional geometry.
An affine space is not modular (for example, if
and
are parallel lines then the formula in the definition of modularity fails). However, it is easy to check that all localizations are modular.
Field extensions and transcendence degree
Let
be a
field extension. The set
becomes a pregeometry if we define
for
. The set
is independent in this pregeometry if and only if it is
algebraically independent over
. The dimension of
coincides with the
transcendence degree
.
In model theory, the case of
being
algebraically closed and
its
prime field is especially important.
While vector spaces are modular and affine spaces are "almost" modular (i.e. everywhere locally modular), algebraically closed fields are examples of the other extremity, not being even locally modular (i.e. none of the localizations is modular).
Strongly minimal sets in model theory
Given a countable first-order language L and an L-
structure M, any definable subset D of M that is
strongly minimal gives rise to a pregeometry on the set D. The closure operator here is given by the algebraic closure in the model-theoretic sense.
A model of a strongly minimal theory is determined up to isomorphism by its dimension as a pregeometry; this fact is used in the proof of
Morley's categoricity theorem.
In minimal sets over
stable theories the independence relation coincides with the notion of forking independence.
References
- H.H. Crapo and G.-C. Rota (1970), On the Foundations of Combinatorial Theory: Combinatorial Geometries. M.I.T. Press, Cambridge, Mass.
- Pillay, Anand (1996), Geometric Stability Theory. Oxford Logic Guides. Oxford University Press.
- Casanovas, Enrique (2008-11-11).
"Pregeometries and minimal types" (PDF).