Theory of Symmetric Lattices [electronic resource] /
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.
Main Authors: | , , |
---|---|
Format: | Texto biblioteca |
Language: | eng |
Published: |
Berlin, Heidelberg : Springer Berlin Heidelberg,
1970
|
Subjects: | Mathematics., Algebra., |
Online Access: | http://dx.doi.org/10.1007/978-3-642-46248-1 |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|