Pregeometry (model theory)
Pregeometry, and in full combinatorial pregeometry, are essentially synonyms for "matroid". They were introduced by Gian-Carlo 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: formula_1, where formula_2 (called the closure map) satisfies the following axioms. For all formula_3 and formula_4: 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 formula_15, formula_16 is  independent over formula_17 if formula_18 for any formula_19. A set formula_20 is a basis for formula_16 over formula_17 if it is independent over formula_17 and formula_24. Since a pregeometry satisfies the Steinitz exchange property all bases are of the same cardinality, hence the definition of the dimension of formula_16 over formula_17 as formula_27 has no ambiguity. The sets formula_28 are independent over formula_29 if formula_30 whenever formula_31 is a finite subset of formula_16. 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 formula_33 is a bijection formula_34 such that formula_35 for any formula_36. A pregeometry formula_33 is said to be homogeneous if for any closed formula_36 and any two elements formula_39 there is an automorphism of formula_33 which maps formula_41 to formula_42 and fixes formula_43 pointwise. The associated geometry and localizations. Given a pregeometry formula_44 its associated geometry (sometimes referred in the literature as the canonical geometry) is the geometry formula_45 where Its easy to see that the associated geometry of a homogeneous pregeometry is homogeneous. Given formula_49 the localization of formula_33 is the geometry formula_51 where formula_52. Types  of pregeometries. Let formula_44 be a pregeometry, then it is said to be: Triviality, modularity and local modularity pass to the associated geometry and are preserved under localization. If formula_33 is a locally modular homogeneous pregeometry and formula_61 then the localization of formula_33 in formula_42 is modular. The geometry formula_33 is modular if and only if whenever formula_65, formula_49, formula_67 and formula_68 then formula_69. Examples. The trivial example. If formula_33 is any set we may define formula_71. This pregeometry is a trivial, homogeneous, locally finite geometry. Vector spaces and projective spaces. Let formula_72 be a field (a division ring actually  suffices) and let formula_73 be a formula_74-dimensional vector space over formula_72. Then formula_73 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. formula_73 is locally finite if and only if formula_72 is finite. formula_73 is not a geometry, as the closure of any nontrivial vector is a subspace of size at least formula_80. The associated geometry of a formula_74-dimensional vector space over formula_72 is the formula_83-dimensional projective space over formula_72. It is easy to see that  this pregeometry is a projective geometry. Affine spaces. Let formula_73 be a formula_74-dimensional affine space over a field formula_72. Given a set define its closure to be its affine hull (i.e. the smallest affine subspace containing it). This forms a homogeneous formula_88-dimensional geometry. An affine space is not modular (for example, if formula_43 and formula_58 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 formula_91 be an algebraically closed field with formula_92, 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). 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.