Overview
- Authors:
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Fumitomo Maeda
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Hiroshima University, Japan
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Shûichirô Maeda
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Ehime University, Japan
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Table of contents (8 chapters)
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- Fumitomo Maeda, Shûichirô Maeda
Pages 1-29
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- Fumitomo Maeda, Shûichirô Maeda
Pages 30-55
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- Fumitomo Maeda, Shûichirô Maeda
Pages 56-71
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- Fumitomo Maeda, Shûichirô Maeda
Pages 72-107
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- Fumitomo Maeda, Shûichirô Maeda
Pages 108-122
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- Fumitomo Maeda, Shûichirô Maeda
Pages 123-135
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- Fumitomo Maeda, Shûichirô Maeda
Pages 136-158
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- Fumitomo Maeda, Shûichirô Maeda
Pages 159-180
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Back Matter
Pages 181-194
About this book
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.
Authors and Affiliations
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Hiroshima University, Japan
Fumitomo Maeda
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Ehime University, Japan
Shûichirô Maeda
