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:
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.
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.
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:
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:
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:
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 .
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.
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.
A pregeometry is said to be modular if it satisfies the following non-obviously equivalent conditions:
Modularity plays an important role in stability theory. One says that a strongly minimal set is modular if the associated pregeometry is modular, for example, and various theorems are proved about this concept. (For example, Zilber proved that any countably categorical strongly minimal set is modular. Hrushovski proved that any modular strongly minimal set with non-trivial associated geometry gives rise to a definable group.) ::: ::: :::