# Theory of Symmetric Lattices

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

Advertisement

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

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.

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

- 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

- Industry Sectors
- Finance, Business & Banking
- Telecommunications