A pregeometry is a set with a finitary closure operator satisfying the Steinitz exchange axiom. That is, is a function from subsets of to subsets of satisfying the following five conditions:

1. Monotonicity: if , then
2. Increasing: for any
3. Idempotent: for any .
4. Finitary: if , then for some finite subset . Equivalently, .
5. Exchange: if and , then .

The first three conditions define a closure operator, and the first four define a finitary closure operator.

Pregeometries are also called matroids (or sometimes, finitary matroids), in other branches of mathematics, including matroid theory, the branch of mathematics which studies pregeometries.

## Examples

If is a vector space over a field , then the linear span yields a pregeometry. The closed sets are the -linear subspaces of .

Similarly, if is a skew field and is a -module, there is a pregeometry whose closed sets are exactly the -submodules of .

If is a field, algebraic closure yields a pregeometry on .

If is an abstract projective plane, there is a pregeometry structure on the points of , in which the closed sets are exactly the empty set, the singletons, the lines, and the entirety of .

If is a finite graph, there is a pregeometry on the set of edges of , in which iff there is a path between the endpoints of consisting of edges in .

If is a pregeometry and , there is an induced pregeometry on , in which for . For example, there is an induced pregeometry structure on any subset of a field.

## [Structure, and equivalent definitions]{#Structure,_and_equivalent_definitions .mw-headline}

Pregeometries have a good deal of additional structure which can be derived from the closure operator. In many cases, one can define pregeometries in terms of this alternative data.

A set is said to be independent if for every . This notion (non-obviously) satisfies the following axioms:

• The empty set is independent.
• Every subset of an independent set is independent.
• A set is independent if and only if every finite subset is independent. (Consequently, the union of any increasing chain of independent sets is independent.)
• For every , the maximal independent subsets of all have the same cardinality.

Conversely, given a set and a class of "independent" sets satisfying the above conditions, there is a unique pregeometry structure on for which the elements of are the independent sets. So this can be used as an alternative definition of pregeometry.

For , the following conditions are equivalent:

• is a maximal independent subset of
• is a minimal subset of such that
• is independent and

If these equivalent conditions are satisfied, is called a basis of (or of ). All bases of have the same cardinality, which is called the rank of .

The rank satisfies the following conditions:

• For any ,
• If , then .
• For any , . Proof: let S be a basis of X cap Y. Let T be a basis of X cup Y extending S.

Conversely, given any function from finite subsets of to natural numbers satisfying these conditions, there is a unique corresponding pregeometry.

A set is closed if it is of the form for some , or equivalently, if . Any intersection of closed sets is closed, and closed sets form a lattice. Closed sets are sometimes called "flats." If is finite, is the unique largest superset having the same rank as .

## Pregeometries in model theory

If is strongly minimal or o-minimal, then algebraic closure yields a pregeometry on the home sort. More generally, if a set is strongly minimal or o-minimal, then algebraic closure on yields a pregeometry.

Generalizing the strongly minimal case, if is a partial type of U-rank 1, in a stable theory, then defines a pregeometry on the set of realizations of .

Even more generally, if is a regular type in a stable theory, there is a natural pregeometry on the set of realizations of . This pregeometry is characterized by the fact that a set is independent if and only if it is independent in the sense of stability theory.

[Maybe this is true: Even more generally, in a stable theory, if is some set of parameters, and is a set of elements such that has weight 1 for every , then there is a pregeometry structure on in which the rank of a finite set is the weight of .]

Outside of a stable context, defines a pregeometry on the home sort in "geometric theories" (by definition). This includes theories such as ACVF and , in which the algebraic closure operator on the home sort agrees with field-theoretic algebraic closure.

## Geometries

A geometry is a pregeometry in which every set of size less than 2 is closed. That is, singletons and the empty set are closed.

To every pregeometry, there is an associated geometry, containing almost all the original data. The association goes a bit like this...

Suppose is a set, , and is an equivalence relation on . Let be the natural map, and let . Then every geometry on yields a pregeometry on , whose closed sets are exactly the sets of the form , for a closed set in .

Conversely, if is a pregeometry, if , if , and if is the equivalence relation on , then there is a unique geometry on yielding the original pregeometry via the above recipe.

The associated geometry has an isomorphic lattice of closed sets.