Pregeometry (model theory)

A combinatorial pregeometry (also known as a finitary matroid), is a second-order structure: ${\displaystyle (X,{\text{cl}})}$, where ${\displaystyle {\text{cl}}:{\mathcal {P}}(X)\to {\mathcal {P}}(X)}$ (called the closure map) satisfies the following axioms. For all ${\displaystyle a,b\in X}$ and ${\displaystyle A,B,C\subseteq X}$:

1. ${\displaystyle {\text{cl}}:({\mathcal {P}}(X),\subseteq )\to ({\mathcal {P}}(X),\subseteq )}$ is a homomorphism in the category of partial orders (monotone increasing), and dominates ${\displaystyle {\text{id}}}$ (I.e. ${\displaystyle A\subseteq B}$ implies ${\displaystyle A\subseteq {\text{cl}}(A)\subseteq {\text{cl}}(B)}$.) and is idempotent.
2. Finite character: For each ${\displaystyle a\in {\text{cl}}(A)}$ there is some finite ${\displaystyle F\subseteq A}$ with ${\displaystyle a\in {\text{cl}}(F)}$.
3. Exchange principle: If ${\displaystyle b\in {\text{cl}}(C\cup \{a\})\smallsetminus {\text{cl}}(C)}$, then ${\displaystyle a\in {\text{cl}}(C\cup \{b\})}$ (and hence by monotonicity and idempotence in fact ${\displaystyle a\in {\text{cl}}(C\cup \{b\})\smallsetminus {\text{cl}}(C)}$).

A geometry is a pregeometry in which the closure of singletons are singletons and the closure of the empty set is the empty set.

Given sets ${\displaystyle A,B\subset S}$, ${\displaystyle A}$ is independent over ${\displaystyle B}$ if ${\displaystyle a\notin {\text{cl}}((A\setminus \{a\})\cup B)}$ for any ${\displaystyle a\in A}$.

A set ${\displaystyle A_{0}\subset A}$ is a basis for ${\displaystyle A}$ over ${\displaystyle B}$ if it is independent over ${\displaystyle B}$ and ${\displaystyle A\subset {\text{cl}}(A_{0}\cup B)}$.

Since a pregeometry satisfies the Steinitz exchange property all bases are of the same cardinality, hence the definition of the dimension of ${\displaystyle A}$ over ${\displaystyle B}$ as ${\displaystyle {\text{dim}}_{B}A=|A_{0}|}$ has no ambiguity.

The sets ${\displaystyle A,B}$ are independent over ${\displaystyle C}$ if ${\displaystyle {\text{dim}}_{B\cup C}=\dim _{C}A'}$ whenever ${\displaystyle A'}$ is a finite subset of ${\displaystyle A}$. Note that this relation is symmetric.

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

A geometry automorphism of a geometry ${\displaystyle S}$ is a bijection ${\displaystyle \sigma :2^{S}\to 2^{S}}$ such that ${\displaystyle \sigma {\text{cl}}(X)={\text{cl}}(\sigma X)}$ for any ${\displaystyle X\subset S}$.

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

Given a pregeometry ${\displaystyle (S,{\text{cl}})}$ its associated geometry (sometimes referred in the literature as the canonical geometry) is the geometry ${\displaystyle (S',{\text{cl}}')}$ where

1. ${\displaystyle S'=\{{\text{cl}}(a)\mid a\in S\setminus {\text{cl}}(\emptyset )\}}$, and
2. For any ${\displaystyle X\subset S}$, ${\displaystyle {\text{cl}}'(\{{\text{cl}}(a)\mid a\in X\})=\{{\text{cl}}(b)\mid b\in {\text{cl}}(X)\}}$

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

Given ${\displaystyle A\subset S}$ the localization of ${\displaystyle S}$ is the geometry ${\displaystyle (S,{\text{cl}}_{A})}$ where ${\displaystyle {\text{cl}}_{A}(X)={\text{cl}}(X\cup A)}$.

Let ${\displaystyle (S,{\text{cl}})}$ be a pregeometry, then it is said to be:

• trivial (or degenerate) if ${\displaystyle {\text{cl}}(X)=\bigcup \{{\text{cl}}(a)\mid a\in X\}}$.
• modular if any two closed finite dimensional sets ${\displaystyle X,Y\subset S}$ satisfy the equation ${\displaystyle {\text{dim}}(X\cup Y)={\text{dim}}(X)+{\text{dim}}(Y)-{\text{dim}}(X\cap Y)}$ (or equivalently that ${\displaystyle X}$ is independent of ${\displaystyle Y}$ over ${\displaystyle X\cap Y}$).
• locally modular if it has a localization at a singleton which is modular.
• (locally) projective if it is non-trivial and (locally) modular.
• 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 ${\displaystyle S}$ is a locally modular homogeneous pregeometry and ${\displaystyle a\in S\setminus {\text{cl}}\emptyset }$ then the localization of ${\displaystyle S}$ in ${\displaystyle b}$ is modular.

The geometry ${\displaystyle S}$ is modular if and only if whenever ${\displaystyle a,b\in S}$, ${\displaystyle A\subset S}$, ${\displaystyle {\text{dim}}\{a,b\}=2}$ and ${\displaystyle {\text{dim}}_{A}\{a,b\}\leq 1}$ then ${\displaystyle ({\text{cl}}\{a,b\}\cap {\text{cl}}(A))\setminus {\text{cl}}\emptyset \neq \emptyset }$.

If ${\displaystyle S}$ is any set we may define ${\displaystyle {\text{cl}}(A)=A}$. This pregeometry is a trivial, homogeneous, locally finite geometry.

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

${\displaystyle V}$ is locally finite if and only if ${\displaystyle F}$ is finite.

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

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

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

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

An affine space is not modular (for example, if ${\displaystyle X}$ and ${\displaystyle Y}$ be parallel lines then the formula in the definition of modularity fails). However, it is easy to check that all localizations are modular.

Let ${\displaystyle k}$ be an algebraically closed field with ${\displaystyle {\text{tr.deg}}(k)\geq \omega }$, 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 modular), algebraically closed fields are examples of the other extremity, not being even locally modular (i.e. none of the localizations is modular).