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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.
List of contents
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 Euclid'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 theLattice 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.
