Theory of Symmetric Lattices
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, i
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Herausgegeben von
J. L. Doob . A. Grothendieck . E. Heinz . F. Hirzebruch E. Hopf· H. Hopf . W. Maak . S. MacLane . W. Magnus M. M. Postnikov . F. K. Schmidt . D. S. Scott . K. Stein
Geschiiftsfuhrende H erausgeber
B. Eckmann und B. L. van der Waerden
F. Maeda . S. Maeda
Theory of Symmetric Lattices
Springer-Verlag Berlin Heidelberg New York 1970
Prof. Dr. Fumitomo Maeda t Late Professor of Mathematics, Hiroshima University
Prof. Dr. ShUichiro Maeda Professor of Mathematics, Ehime University
Geschiiftsfiihrende Herausgeber:
Prof. Dr. B. Eckmann Eidgenossische Technische Hochschule Ziirich
Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universitiit Ziirich
AMS Subject Classifications (1970): Primary06A30· Secondary SODOS, 46ESO, 46LlO ISBN-13: 978-3-642-46250-4 e-ISBN-13: 978-3-642-46248-1 DOl: 10.1007/978-3-642-46248-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re·use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.
Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1970. Library of Congress Catalog Card Number 73-128138 Softcover reprint of the hardcover 1st edition 1970
Title No. 5156
Preface 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. Continuous 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-symmetric, 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. Furthermore we can show that this lattice has a modular extension. On the other hand, an M-symmetric lattice with a modular extension was introduced by R. L. Wilcox, and it bears his name. An affine matroid l\lttice is an atomistic Wilcox lattice. In general Wilcox lattice, we introduce the concept of point-free parallelism to extend the theory of parallelism in the atomistic case. We may say that matroid latt
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