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, there 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.

Definitions

Pregeometries and geometries

A combinatorial pregeometry (also known as a finitary matroid), is a second-order structure: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (X,\text{cl})\, , where Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{cl}:\mathcal{P}(X)\to\mathcal{P}(X)\,

(called the closure map) satisfies the following axioms. For all $a, b\in X\,$ and $A, B,C\subseteq X\,$:

1. Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{cl}:(\mathcal{P}(X),\subseteq)\to(\mathcal{P}(X),\subseteq)\,

is an homomorphism in the category of partial orders (monotone increasing), and dominates Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{id}\,
(I.e. $A\subseteq B\,$ implies Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): A\subseteq\text{cl}(A)\subseteq\text{cl}(B)\,

.) and is idempotent.

1. Finite character: For each Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a\in\text{cl}(A)\,

there is some finite $F\subseteq A\,$ with Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a\in\text{cl}(F)\,

.

1. Exchange principle: If Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): b\in\text{cl}(C\cup\{a\})\smallsetminus\text{cl}(C)\,

, then Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a\in\text{cl}(C\cup\{b\})

(and hence by monotonicity and idempotence in fact Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a\in\text{cl}(C\cup\{b\})\smallsetminus\text{cl}(C)\,

).

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 $A,B\subset S$, $A$ is independent over $B$ if Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a\notin \text{cl}((A\setminus\{a\})\cup B)

for any $a\in A$.

A set $A_0 \subset A$ is a basis for $A$ over $B$ if it is independent over $B$ and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): A\subset \text{cl}(A_0\cup B) .

Since a pregeometry satisfies the Steinitz exchange property all bases are of the same cardinality, hence the definition of the dimension of $A$ over $B$ as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{dim}_B A = |A_0|

has no ambiguity.

The sets $A,B$ are independent over $C$ if Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{dim}_{B\cup C} = \dim_C A'

whenever $A'$ is a finite subset of $A$. Note that this relation is symmetric.

In minimal sets over stable theories the independence relation coincides with the notion of forking independence.

Geometry automorphism

A geometry automorphism of a geometry $S$ is a bijection $\sigma:2^S\to 2^S$ such that Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sigma\text{cl}(X)=\text{cl}(X)\sigma

for any $X\subset S$.

A pregeometry $S$ is said to be homogeneous if for any closed $X\subset S$ and any two elements $a,b\in S\setminus X$ there is an automorphism of $S$ which maps $a$ to $b$ and fixes $X$ pointwise.

The associated geometry and localizations

Given a pregeometry Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (S,\text{cl})

its associated geometry (sometimes referred in the literature as the canonical geometry) is the geometry Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (S',\text{cl}')
where

1. Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): S'=\{\text{cl}(a)\mid a\in S\setminus \text{cl} (\emptyset)\}

, and

1. For any $X\subset S$, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{cl}'(\{\text{cl}(a)\mid a\in X = \{\text{cl}'(b)\mid b\in\text{cl}X\}

Its easy to see that the associated geometry of a homogeneous pregeometry is homogeneous.

Given $A\subset S$ the localization of $S$ is the geometry Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (S,\text{cl}_A)

where Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{cl}_A(X)=\text{cl}(X\cup A)

.

Types of pregeometries

Let Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (S,\text{cl})

be a pregeometry, then it is said to be:

• trivial (or degenerate) if Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{cl}(X)=\bigcup\{\text{cl}(a)\mid a\in X\}

,

• modular if any two closed finite dimensional sets $X,Y\subset S$ satisfy the equation Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{dim}(X\cup Y) = \text{dim}(X) + \text{dim}(Y) - \text{dim}(X\cap Y)

(or equivalently that $X$ is independent of $Y$ over $X\cap Y$.

• 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 $S$ is a locally modular homogeneous pregeometry and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a\in S\setminus\text{cl}\emptyset

then the localization of $S$ in $b$ is modular.

The geometry $S$ is modular if and only if whnever $a,b\in S$, $A\subset S$, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{dim}\{a,b\}=2

and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{dim}_A\{a,b\} \le 1
then Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (\text{cl}\{a,b\}\cap\text{cl}(A))\setminus\text{cl}\emptyset\ne\emptyset

.

Examples

The trivial example

If $S$ is any set we may define Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{cl}(A)=A . This pregeometry is a trivial, homogeneous, locally finite geometry.

Vector spaces and projective spaces

Let $F$ be a field (a division ring actually suffices) and let $V$ be a $\kappa$-dimensional vector space over $F$. Then $V$ 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.

$V$ is locally finite if and only if $F$ is finite.

$V$ is not a geometry, as the closure of any nontrivial vector is a subspace of size at least $2$.

The associated geometry of a $\kappa$-dimensional vector space over $F$ is the $(\kappa-1)$-dimensional projective space over $F$. It is easy to see that this pregeometry is a projective geometry.

Affine spaces

Let $V$ be a $\kappa$-dimensional affine space over a field $F$. Given a set define its closure to be its affine hull (i.e. the smallest affine subspace containing it).

This forms a homogeneous $(\kappa+1)$-dimensional geometry.

An affine space is not modular (for example, if $X$ and $Y$ 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 $k$ be an algebraically closed field with Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \text{tr.deg}(k)\ge\omega , 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).

References

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.