Pregeometry

A pregeometry (also called abstract pregeometry) consists of a set <math> X <math>, and a function <math> cl <math> (called clausure) which maps subsets of <math> X <math> to subsets of <math> X <math>, that is: <math> cl : P(X) --> P(X) <math>, and satisfies the following conditions, for all <math> a, b \in X <math> and all <math> Y, Z \subseteq X <math>:

1. <math> Y \subseteq cl(Y) <math>.
2. If Y <math>\subseteq<math> Z, then <math> cl(Y) \subseteq cl(Z) <math>.
3. <math> cl(cl(Y)) = cl(Y) <math>.
4. (finite character) If <math> a \in cl(Y) <math>, then there is a finite subset of Y, Y', such that <math> a \in cl(Y') <math>.
5. (exchange principle) If <math> a \in cl( Y b ) \smallsetminus cl(Y) <math>, then <math> b \in cl( Y a ) <math>. [here <math> Ya <math> is <math> Y \cup \{ a \} <math>, similar for <math> Yb <math>].

A geometry is a pregeometry such that <math> cl(\{ a \}) = \{ a \} <math> for all <math> a \in X <math>.

For example, let <math> V <math> be a vector space over a field, and, for <math> Y \subseteq V <math>, define <math> cl(Y) <math> to be the span of <math> Y <math>, that is, the set of linear combinations of elements of <math> Y <math>. Then the pair <math> (V, cl) <math> is a pregeometry, as it is easy to see.

In contrast, if <math> X <math> is a topological space and we define <math> cl <math> to be the topological-closure function, then the pair <math> (X, cl) <math> will not neccesarily be a pregeometry, as the finite character condition (4) may fail.

It turns out that many fundamental concepts of linear algebra --closure, independence, subspace, basis, dimension-- are preserved in the framework of abstract geometries.

Let <math> (X, cl) <math> be a pregeometry, and <math> B, Y <math> be subsets of <math> X <math>. We will say that <math> Y <math> is closed if <math> cl(Y) = Y <math>, and that <math> B <math> generates <math> Y <math> if <math> Y = cl(B) <math>. Also we will say that <math> B <math> is independent if no proper subset generates <math> cl(B) <math>, that is, for all <math> B' \subsetneq B <math>, <math> cl(B') \subsetneq cl(B) <math>.

If <math> B <math> is independent and generates <math> Y <math>, then we will say that <math> B <math> is a base for <math> Y <math>. Equivalently, a base for <math> Y <math> is a minimal <math> Y <math>-generating set, or (by Zorn's Lemma) a maximal independent subset of <math> Y <math>.