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.
Pregeometries and geometries
A combinatorial pregeometry (also known as a finitary matroid), is a second-order structure: , where (called the closure map) satisfies the following axioms. For all and :
- is an homomorphism in the category of partial orders (monotone increasing), and dominates (I.e. implies .) and is idempotent.
- Finite character: For each there is some finite with .
- Exchange principle: If , then (and hence by monotonicity and idempotence in fact ).
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 for over 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.
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 (sometimes referred in the literature as the canonical geometry) is the geometry where
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 (or degenerate) if ,
- 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
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.
If is a locally modular homogeneous pregeometry and then the localization of in is modular.
The geometry is modular if and only if whnever , , and then .
The trivial example
If is any set we may define . 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 -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.
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 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 be an algebraically closed field with , 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).
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.