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. It turns out that many fundamental concepts of linear algebra – closure, independence, subspace, basis, dimension – are available in the general framework of pregeometries. 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. Motivation If is a vector space over some field and , we define to be the set of all linear combinations of vectors from , also known as the span of . Then we have and and . The Steinitz exchange lemma is equivalent to the statement: if , then The linear algebra concepts of independent set, generating set, basis and dimension can all be expressed using the -operator alone. A pregeometry is an abstraction of this situation: we start with an arbitrary set and an arbitrary operator which assigns to each subset of a subset of , satisfying the properties above. Then we can define the "linear algebra" concepts also in this more general setting. This generalized notion of dimension is very useful in model theory, where in certain situation one can argue as follows: two models with the same cardinality must have the same dimension and two models with the same dimension must be isomorphic. Definitions Pregeometries and geometries A combinatorial pregeometry (also known as a finitary matroid) is a pair , where is a set and (called the closure map) satisfies the following axioms. For all and : is monotone increasing and dominates (i.e. implies ) and is idempotent (i.e. ) Finite character: For each there is some finite with . Exchange principle: If , then (and hence by monotonicity and idempotence in fact ). Sets of the form for some are called closed. It is then clear that finite intersections of closed sets are closed and that is the smallest closed set containing . A geometry is a pregeometry in which the closure of singletons are