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 cacaphonous" 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. The study of how pregeometries, geometries, and abstract closure operators influence the structure of first-order models is called geometric stability theory.

Definition

A combinatorial pregeometry, also known as a finitary matroid, consists of a set $X$, and a function $cl$ (called "closure") which maps subsets of $X$ to subsets of $X$, that is: $cl\; :\; P(X)\; ightarrow\; P(X)$, and satisfies the following conditions, for all $a,\; b\; in\; X$ and all $Y,\; Z\; subseteq\; X$:

# $Y\; subseteq\; cl(Y)$.
# If Y $subseteq$ Z, then $cl(Y)\; subseteq\; cl(Z)$.
# $cl(cl(Y))\; =\; cl(Y)$.
# (finite character) If $a\; in\; cl(Y)$, then there is a finite subset of Y, Y', such that $a\; in\; cl(Y\text{'})$.
# (exchange principle) If $a\; in\; cl(\; Y\; b\; )\; smallsetminus\; cl(Y)$, then $b\; in\; cl(\; Y\; a\; )$. [here $Ya$ is $Y\; cup\; \{\; a\; \}$, similar for $Yb$] .

A geometry is a pregeometry such that $cl(\{\; a\; \})\; =\; \{\; a\; \}$ for all $a\; in\; X$ and also $cl(emptyset)\; =\; emptyset$.

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

Let $(X,\; cl)$ be a pregeometry, and $B,\; Y$ be subsets of $X$. We will say that $Y$ is "closed" if $cl(Y)\; =\; Y$, and that $B$ "generates" $Y$ if $Y\; =\; cl(B)$. Also we will say that $B$ is independent if no proper subset generates $cl(B)$, that is, for all $B\text{'}\; subsetneq\; B$, $cl(B\text{'})\; subsetneq\; cl(B)$.

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

Examples

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

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

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.