COMPATIBILITY OF EXTENSIONS OF A COMBINATORIAL GEOMETRY
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Date
2016-09-29
Authors
Cheung, Alan
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
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.
Description
Keywords
combinatorial geometries, elementary extensions, strong maps, orthogonality, compatibility theorems, quotient bundles, lift and drop sequences, linear subclass generating sequence, weak order