COMPATIBILITY OF EXTENSIONS OF A COMBINATORIAL GEOMETRY

Abstract

Two extensions of a geometry are compatible with each other if they have a common extension. If the given extensions are elementary, their compatibility can be intrinsically described in terms of their corresponding linear subclasses. Certain adjointness relation between an extension of a geometry and the geometry itself is also discussed. Any extension of a geometry G by a geometry F determines and is determined by a unique quotient bundle on G indexed by F. As a study of the compatibility among given quotients of a geometry, we look at the possibility of completing to F-bundles a family of quotients indexed by a set I of flats of F. If the indexing geometry F is free and if the set I is a Boolean subalgebra or a sublattice of F, for any family Q(I) of quotients of a geometry G, there is a canonical construction which determines its completability and at the same time produces the extremal completion if it is a partial bundle. Geometries studied in this dissertation are furnished with the weak order. Almost invariably, the Higgs' lift construction, in a somewhat generalized sense, constitutes a convenient and indispensable means in various of the extremal constructions.