Automorphisms, flats and erections of pregeometries

We characterize pregeometries in terms of their flats - giving necessary and sufficient conditions for a collection of sets to be the flats of a pregeometry. We begin a systematic study of automorphisms of pregeometries; arguing geometrically, we obtain the (known) factorization of a pregeometry into component pregeometries, and express the automorphism group of the pregeometry in terms of the automorphism groups of its components. We use automorphisms of pregeometries and their characterization above by flats in the following three ways: 1. All erections of any pregeometry are constructed and their automorphism group determined. 2. We indicate a process for obtaining all pregeometries on a set from the geometries on smaller sets and their automorphism groups. 3. We begin the study of the transitivity of the automorphism group of a pregeometry on the flats and the elements of the pregeometry.