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AuthorMaeda, Fumitomo. author
TitleTheory of Symmetric Lattices [electronic resource] / by Fumitomo Maeda, Shรปichirรด Maeda
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1970
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Descript XII, 194 p. online resource


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. Continuยญ 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-symยญ 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. Furtherยญ more we can show that this lattice has a modular extension


I Symmetric Lattices and Basic Properties of Lattices -- 1. Modularity in Lattices -- 2. Semi-orthogonality in Lattices -- 3. Semi-orthogonality in ?-Symmetric Lattices -- 4. Distributivity and the Center of a Lattice -- 5. Centers of Complete Lattices -- 6. Perspectivity and Projectivity in Lattices -- II Atomistic Lattices and the Covering Property -- 7. The Covering Property in Atomistic Lattices -- 8. Atomistic Lattices with the Covering Property -- 9. Finite-modular AC-lattices -- 10. Distributivity and Perspectivity in Atomistic Lattices -- 11. Perspectivity in AC-Lattices -- 12. Completion by Cuts -- III Matroid Lattices -- 13. Perspectivity and Irreducible Decompositions of Matroid Lattices -- 14. Modularity in Matroid Lattices -- 15. Atom Spaces of Atomistic Lattices -- 16. Projective Spaces and Modular Matroid Lattices -- IV Parallelism in Symmetric Lattices -- 17. Parallelism in Lattices -- 18. Incomplete Elements in Affine Matroid Lattices -- 19. Modular Contractions and Modular Extensions of Affine Matroid Lattices -- 20. Atomistic Wilcox Lattices -- 21. Singular Elements in Atomistic Wilcox Lattices -- 22. Affine Matroid Lattices Satisfying Euclidโ{128}{153}s Strong Parallel Axiom -- V Point-free Parallelism in Symmetric Lattices -- 23. Point-free Parallelism in Lattices -- 24. Point-free Parallelism in Wilcox Lattices -- 25. Uniqueness of the Modular Extension of a Wilcox Lattice -- 26. Modular Contractions and Modular Centers of Wilcox Lattices -- VI Atomistic Symmetric Lattices with Duality -- 27. Modularity in DAC-lattices -- 28. Complete DAC-lattices -- 29. Orthocomplemented Lattices and Orthomodular Lattices -- 30. Orthocomplemented AC-lattices -- VII Atomistic Lattices of Subspaces of Vector Spaces -- 31. The Lattice of Closed Subspaces of a Locally Convex Space -- 32. Modular Pairs in the Lattice of Closed Subspaces -- 33. Pairs of Dual Spaces -- 34. Vector Spaces with Hermitian Forms -- VIII Orthomodular Symmetric Lattices -- 35. Relatively Complemented Symmetric Lattices with Duality -- 36. Commutativity in Orthomodular Lattices -- 37. Lattices of Projections of Baer *-semigroups -- 38. Modular Pairs in Lattices of Projections -- Supplement -- List of Special Symbols

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