Theory of Symmetric Lattices
Maeda, Fumitomo, Maeda, Shuichiro
Produktnummer:
183eb78e50722243e398342476a18f148c
Autor: | Maeda, Fumitomo Maeda, Shuichiro |
---|---|
Themengebiete: | Finite Lattice Lattices Verband duality eXist form geometry matroid modularity |
Veröffentlichungsdatum: | 17.03.2012 |
EAN: | 9783642462504 |
Sprache: | Englisch |
Seitenzahl: | 194 |
Produktart: | Kartoniert / Broschiert |
Verlag: | Springer Berlin |
Produktinformationen "Theory of Symmetric Lattices"
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.

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