Pregeometry

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


 * 1) Monotonicity: if $$X \subseteq Y \subseteq S$$, then $$cl(X) \subseteq cl(Y)$$
 * 2) Increasing: $$cl(X) \supseteq X$$ for any $$X \subseteq S$$
 * 3) Idempotent: $$cl(cl(X)) = cl(X)$$ for any $$X \subseteq S$$.
 * 4) Finitary: if $$x \in cl(X)$$, then $$x \in cl(X')$$ for some finite subset $$X' \subset X$$. Equivalently, $$cl(X) = \bigcup_{X' \subset_f X} cl(X')$$.
 * 5) Exchange: if $$x \in cl(X \cup \{y\})$$ and $$x \notin cl(X)$$, then $$y \in cl(X \cup \{x\})$$.

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

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

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

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

If $$G$$ is a finite graph, there is a pregeometry on the set of edges of $$G$$, in which $$e \in cl(X)$$ iff there is a path between the endpoints of $$e$$ consisting of edges in $$X$$.

If $$(S,cl)$$ is a pregeometry and $$S' \subset S$$, there is an induced pregeometry $$(S',cl')$$ on $$S'$$, in which $$cl'(X) = cl(X) \cap S'$$ for $$X \subset S'$$. For example, there is an induced pregeometry structure on any subset of a field.

Structure, and equivalent definitions
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 $$I \subset S$$ is said to be independent if $$i \notin cl(I \setminus \{i\})$$ for every $$i \in I$$. 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 $$X \subset S$$, the maximal independent subsets of $$X$$ all have the same cardinality.

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

For $$B \subset X \subset S$$, the following conditions are equivalent:


 * $$B$$ is a maximal independent subset of $$S$$
 * $$B$$ is a minimal subset of $$S$$ such that $$X \subset cl(B)$$
 * $$B$$ is independent and $$X \subset cl(B)$$

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

The rank satisfies the following conditions:


 * For any $$X$$, $$0 \le rank(X) \le |X|$$
 * If $$X \subset Y$$, then $$rank(X) \le rank(Y)$$.
 * For any $$X, Y$$, $$rank(X \cup Y) + rank(X \cap Y) \le rank(X) + rank(Y)$$. 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 $$X$$ to natural numbers satisfying these conditions, there is a unique corresponding pregeometry.

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

Pregeometries in model theory
If $$T$$ is strongly minimal or o-minimal, then algebraic closure yields a pregeometry on the home sort. More generally, if a set $$D$$ is strongly minimal or o-minimal, then algebraic closure on $$D$$ yields a pregeometry.

Generalizing the strongly minimal case, if $$p$$ is a partial type of U-rank 1, in a stable theory, then $$\operatorname{acl}$$ defines a pregeometry on the set of realizations of $$p$$.

Even more generally, if $$p$$ is a regular type in a stable theory, there is a natural pregeometry on the set of realizations of $$p$$. 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 $$A$$ is some set of parameters, and $$S$$ is a set of elements such that $$\operatorname{tp}(s/A)$$ has weight 1 for every $$s \in S$$, then there is a pregeometry structure on $$S$$ in which the rank of a finite set $$S_0 \subset S$$ is the weight of $$\operatorname{tp}(S_0/A)$$.]

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

Geometries
A geometry is a pregeometry $$(S,cl)$$ 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 $$S$$ is a set, $$S' \subset S$$, and $$E$$ is an equivalence relation on $$S'$$. Let $$f : S' \to S'/E$$ be the natural map, and let $$Z = S \setminus S'$$. Then every geometry $$cl(-)$$ on $$S'/E$$ yields a pregeometry on $$S$$, whose closed sets are exactly the sets of the form $$f^{-1}(C) \cup Z$$, for $$C$$ a closed set in $$S'/E$$.

Conversely, if $$(S,cl)$$ is a pregeometry, if $$Z = cl(\emptyset)$$, if $$S' = S \setminus Z$$, and if $$E$$ is the equivalence relation $$xEy \iff cl(x) = cl(y)$$ on $$S'$$, then there is a unique geometry on $$S'/E$$ yielding the original pregeometry via the above recipe.

The associated geometry has an isomorphic lattice of closed sets.

Modular (pre)geometries
A pregeometry is said to be modular if it satisfies the following non-obviously equivalent conditions:


 * The lattice of closed sets is modular.
 * The lattice of finite rank closed sets is modular.
 * If $$X$$ and $$Y$$ are closed sets of finite rank, then $$rank(X \cup Y) + rank(X \cap Y) = rank(X) + rank(Y)$$.

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.)