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Theory of Symmetric Lattices

  • Fumitomo Maeda
  • Shûichirô Maeda

Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 173)

Table of contents

  1. Front Matter
    Pages I-XI
  2. Fumitomo Maeda, Shûichirô Maeda
    Pages 1-29
  3. Fumitomo Maeda, Shûichirô Maeda
    Pages 30-55
  4. Fumitomo Maeda, Shûichirô Maeda
    Pages 56-71
  5. Fumitomo Maeda, Shûichirô Maeda
    Pages 72-107
  6. Fumitomo Maeda, Shûichirô Maeda
    Pages 108-122
  7. Fumitomo Maeda, Shûichirô Maeda
    Pages 123-135
  8. Fumitomo Maeda, Shûichirô Maeda
    Pages 136-158
  9. Fumitomo Maeda, Shûichirô Maeda
    Pages 159-180
  10. Back Matter
    Pages 181-194

About this book

Introduction

Of central importance in this book is the concept of modularity in lattices. A lattice is said to be modular if every pair of its elements is a modular pair. The properties of modular lattices have been carefully investigated by numerous mathematicians, including 1. von Neumann who introduced the important study of continuous geometry. Continu­ ous geometry is a generalization of projective geometry; the latter is atomistic and discrete dimensional while the former may include a continuous dimensional part. Meanwhile there are many non-modular lattices. Among these there exist some lattices wherein modularity is symmetric, that is, if a pair (a,b) is modular then so is (b,a). These lattices are said to be M-sym­ metric, and their study forms an extension of the theory of modular lattices. An important example of an M-symmetric lattice arises from affine geometry. Here the lattice of affine sets is upper continuous, atomistic, and has the covering property. Such a lattice, called a matroid lattice, can be shown to be M-symmetric. We have a deep theory of parallelism in an affine matroid lattice, a special kind of matroid lattice. Further­ more we can show that this lattice has a modular extension.

Keywords

Finite Lattice Lattices Verband duality eXist form geometry matroid modularity parallelism projective geometry semigroup sets

Authors and affiliations

  • Fumitomo Maeda
    • 1
  • Shûichirô Maeda
    • 2
  1. 1.Hiroshima UniversityJapan
  2. 2.Ehime UniversityJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-46248-1
  • Copyright Information Springer-Verlag Berlin Heidelberg 1970
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-46250-4
  • Online ISBN 978-3-642-46248-1
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site