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.

Bibliographic Details

Main Authors:	Maeda, Fumitomo. author., Maeda, Shûichirô. author., SpringerLink (Online service)
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
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