Theory of Symmetric Lattices

By Fumitomo Maeda, Shuichiro Maeda, Shuichiro Maeda Fumitomo Maeda, et al.

Book Overview

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.

• Format: Paperback
• Language: English
• ISBN: 3642462502
• ISBN13: 9783642462504
• Release Date: March 2012
• Publisher: Springer
• Length: 194 Pages
• Weight: 0.67 lbs.
• Dimensions: 0.5" x 6.1" x 9.2"