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.

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 ${\displaystyle A,B,C\subseteq X\,}$:

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