Truncations of principal geometries

Abstract

We investigate the class of principal pregeometries (free simplicial geometries with spanning simplex) which form an important subclass of the class of transversal pregeometries (free simplicial geometries). We give a coordinate-free method for imbedding a transversal pregeometry on a simplex as a free simplicial pregeometry which makes use only of the set-theoretic properties of a presentation of the transversal pregeometry. We introduce the notion of an (r, k)-principal set as a generalization of principal basis and prove the collection of (r, k)-principal sets of a rank k pregeometry, if non-empty, are the bases of another pregeometry whose structure is determined. An algorithm for constructing principal sets is given. We then characterize truncations of principal geometries in terms of the existence of a principal set. We do this by erecting a given pregeometry to a free simplicial pregeometry with spanning simplex. The erection is the freest of all erections of the given pregeometry.