# Theory of Symmetric Lattices

• Fumitomo Maeda
• Shûichirô Maeda
Book

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

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

### 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
• Print ISBN 978-3-642-46250-4
• Online ISBN 978-3-642-46248-1
• Series Print ISSN 0072-7830