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. In the branch of mathematical logic called model theory, infinite finitary matroids, there called "pregeometries", 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.
Definitions
Pregeometries and geometries
A combinatorial pregeometry, is a second-order structure:, where satisfies the following axioms. For all and :
is a homomorphism in the category of partial orders, and dominates and is idempotent.
Finite character: For each there is some finite with.
Exchange principle: If, then .
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. A set is a basis forover if it is independent over and. Since a pregeometry satisfies the Steinitz exchange property all bases are of the same cardinality, hence the definition of the dimension of over as has no ambiguity. The sets are independent over if whenever is a finite subset of. Note that this relation is symmetric. In minimal sets over stable theories the independence relation coincides with the notion of forking independence.
Geometry automorphism
A geometry automorphism of a geometry 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 is the geometry where
, and
For any,
Its easy to see that the associated geometry of a homogeneous pregeometry is homogeneous. Given the localization of is the geometry where.
Types of pregeometries
Let be a pregeometry, then it is said to be:
trivial if.
modular if any two closed finite dimensional sets satisfy the equation .
locally modular if it has a localization at a singleton which is modular.
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. This pregeometry is a trivial, homogeneous, locally finite geometry.
Let be a field and let be a -dimensional vector space over. Then is a pregeometry where closures of sets are defined to be their span. 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. This forms a homogeneous -dimensional geometry. An affine space is not modular. However, it is easy to check that all localizations are modular.
