Pregeometry (model theory)
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 , 

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. Matroid theory

Model theory