# Theory of Symmetric Lattices

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

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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
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