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.
to be the set of all linear combinations of vectors from
The Steinitz exchange lemma is equivalent to the statement: if
The linear algebra concepts of independent set, generating set, basis and dimension can all be expressed using the
A pregeometry is an abstraction of this situation: we start with an arbitrary set
Then we can define the "linear algebra" concepts also in this more general setting.
A combinatorial pregeometry (also known as a finitary matroid) is a pair
(called the closure map) satisfies the following axioms.
A geometry is a pregeometry in which the closure of singletons are singletons and the closure of the empty set is the empty set.
Since a pregeometry satisfies the Steinitz exchange property all bases are of the same cardinality, hence we may define the dimension of
is defined to be the dimension over the empty set.
is a locally modular homogeneous pregeometry and
This pregeometry is a trivial, homogeneous, locally finite geometry.
be a field (a division ring actually suffices) and let
is a pregeometry where closures of sets are defined to be their span.
The closed sets are the linear subspaces of
Vector spaces are considered to be the prototypical example of modularity.
is not a geometry, as the closure of any nontrivial vector is a subspace of size at least
It is easy to see that this pregeometry is a projective geometry.
An affine space is not modular (for example, if
are parallel lines then the formula in the definition of modularity fails).
However, it is easy to check that all localizations are modular.
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.