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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.
| Autorius: | Shuichiro Maeda, Fumitomo Maeda, |
| Serija: | Grundlehren der mathematischen Wissenschaften |
| Leidėjas: | Springer Berlin Heidelberg |
| Išleidimo metai: | 2012 |
| Knygos puslapių skaičius: | 212 |
| ISBN-10: | 3642462502 |
| ISBN-13: | 9783642462504 |
| Formatas: | 235 x 155 x 12 mm. Knyga minkštu viršeliu |
| Kalba: | Anglų |
Parašykite atsiliepimą apie „Theory of Symmetric Lattices“
