# Pregeometry (model theory)

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.

In the branch of mathematical logic called model theory, infinite finitary matroids, they are 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.

## Definitions

### Pregeometries and geometries

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

Since a pregeometry satisfies the Steinitz exchange property all bases are of the same cardinality, hence the definition of the dimension of $A$ over $B$ as ${\text{dim}}_{B}A=|A_{0}|$ has no ambiguity.

In minimal sets over stable theories the independence relation coincides with the notion of forking independence.

### the associated geometry and localizations

Given a pregeometry $(S,{\text{cl}})$ its associated geometry (sometimes referred in the literature as the canonical geometry) is the geometry $(S',{\text{cl}}')$ where

Its easy to see that the associated geometry of a homogeneous pregeometry is homogeneous.

### types of pregeometries

4. (locally) projective if it is non-trivial and (locally) modular 5. locally finite if closures of finite sets are finite

Triviality, modularity and local modularity pass to the associated geometry and are preserved under localization.

## Examples

### The trivial example

If $S$ is any set we may define ${\text{cl}}(A)=A$ . This pregeometry is a trivial, homogeneous, locally finite geometry.

### Vector spaces and projective spaces

This pregeometry is homogeneous and modular. Vector spaces are considered to be the prototypical example of modularity.

$V$ is not a geometry, as the closure of any nontrivial vector is a subspace of size at least $2$ .

### Affine spaces

This forms a homogeneous $(\kappa +1)$ -dimensional geometry.

An affine space is not modular (for example, if $X$ and $Y$ be parallel lines then the formula in the definition of modularity fails). However, it is easy to check that all localizations are modular.

### Algebraically closed fields

Let $k$ be an algebraically closed field with ${\text{tr.deg}}(k)\geq \omega$ , and define the closure of a set to be its algebraic closure.

While vector spaces are modular and affine spaces are "almost" modular (i.e. everywhere locally modulare), algebraically closed fields are examples of the other extremity, not being even locally modular (i.e. none of the localizations is modular).