Theory Of Symmetric Lattices (grundlehren Der Mathematischen Wissenschaften)

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. ContinuƯ 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-symƯ 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. FurtherƯ more we can show that this lattice has a modular extension Front Matter....Pages I-XI Symmetric Lattices and Basic Properties of Lattices....Pages 1-29 Atomistic Lattices and the Covering Property....Pages 30-55 Matroid Lattices....Pages 56-71 Parallelism in Symmetric Lattices....Pages 72-107 Point-free Parallelism in Symmetric Lattices....Pages 108-122 Atomistic Symmetric Lattices with Duality....Pages 123-135 Atomistic Lattices of Subspaces of Vector Spaces....Pages 136-158 Orthomodular Symmetric Lattices....Pages 159-180 Back Matter....Pages 181-194