A Bundle Representation for Continuous Geometries

John Harding, Melvin F. Janowitz
1997 Advances in Applied Mathematics  
We show that a reducible continuous geometry can be represented as the continuous sections of a bundle of irreducible continuous geometries. We relate this bundle representation to the Pierce sheaf of the continuous geometry and to the subdirect product representation Ä 4 Ž . the family of dimension functions D : J g Y and in fact D arJ s J J Ž .Ž . w x D a J l Z 5, Hilfsatz 3.3, p. 124 . In particular, we shall need the fact that each D is a positive modular valuation on LrJ, so it induces a
more » ... tric J on LrJ. Ž . For elements a, b g L we will often use the notation D a to denote J Ž . Ž . D arJ and d a, b to denote the associated distance between the J J Ž . elements arJ and brJ. It will be helpful to note that since d a, b s J
doi:10.1006/aama.1997.0548 fatcat:o3gqgb762bhqzk5tdahldwvbcu