{"product_id":"9783642462504","title":"Grundlehren der mathematischen Wissenschaften","description":"\u003ch1\u003eGrundlehren der mathematischen Wissenschaften\u003c\/h1\u003e \u003ch2\u003eMaeda, Fumitomo; Maeda, Shuichiro\u003c\/h2\u003e \u003cp\u003eOf 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.\u003c\/p\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Springer\u003c\/p\u003e \u003cp\u003ePublication Date: 2012-03-17\u003c\/p\u003e \u003cp\u003eFormat: Paperback\u003c\/p\u003e \u003cp\u003eISBN-13: 9783642462504\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-3-642-46248-1\u003c\/p\u003e \u003cp\u003eDimensions: 235cm x155cm\u003c\/p\u003e \u003cp\u003ePages: 194\u003c\/p\u003e ","brand":"Springer Berlin Heidelberg","offers":[{"title":"Default Title","offer_id":45379228074124,"sku":"9783642462504","price":49.49,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9783642462504.jpg?v=1775731762","url":"https:\/\/fh90cf-fv.myshopify.com\/products\/9783642462504","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}