Pregeometry, and in full combinatorial pregeometry, are essentially synonyms for "matroid". They were introduced by G.-C. 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.
Contents
- Pregeometries and geometries
- Independence bases and dimension
- Geometry automorphism
- The associated geometry and localizations
- Types of pregeometries
- The trivial example
- Vector spaces and projective spaces
- Affine spaces
- Algebraically closed fields
- References
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.
It turns out that many fundamental concepts of linear algebra – closure, independence, subspace, basis, dimension – are preserved in the framework of abstract geometries.
The study of how pregeometries, geometries, and abstract closure operators influence the structure of first-order models is called geometric stability theory.
Pregeometries and geometries
A combinatorial pregeometry (also known as a finitary matroid), is a second-order structure:
-
cl : ( P ( X ) , ⊆ ) → ( P ( X ) , ⊆ ) is an homomorphism in the category of partial orders (monotone increasing), and dominates id (I.e. A ⊆ B implies A ⊆ cl ( A ) ⊆ cl ( B ) .) and is idempotent. - Finite character: For each
a ∈ cl ( A ) there is some finite F ⊆ A with a ∈ cl ( F ) . - Exchange principle: If
b ∈ cl ( C ∪ { a } ) ∖ cl ( C ) , then a ∈ cl ( C ∪ { b } ) (and hence by monotonicity and idempotence in facta ∈ cl ( C ∪ { b } ) ∖ cl ( C ) ).
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
A set
Since a pregeometry satisfies the Steinitz exchange property all bases are of the same cardinality, hence the definition of the dimension of
The sets
In minimal sets over stable theories the independence relation coincides with the notion of forking independence.
Geometry automorphism
A geometry automorphism of a geometry
A pregeometry
The associated geometry and localizations
Given a pregeometry
-
S ′ = { cl ( a ) ∣ a ∈ S ∖ cl ( ∅ ) } , and - For any
X ⊂ S ,cl ′ ( { cl ( a ) ∣ a ∈ X } ) = { cl ( b ) ∣ b ∈ cl X }
Its easy to see that the associated geometry of a homogeneous pregeometry is homogeneous.
Given
Types of pregeometries
Let
Triviality, modularity and local modularity pass to the associated geometry and are preserved under localization.
If
The geometry
The trivial example
If
Vector spaces and projective spaces
Let
This pregeometry is homogeneous and modular. Vector spaces are considered to be the prototypical example of modularity.
The associated geometry of a
Affine spaces
Let
This forms a homogeneous
An affine space is not modular (for example, if
Algebraically closed fields
Let
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).