Theory of Symmetric Lattices, Softcover reprint of the original 1st ed. 1970 Grundlehren der mathematischen Wissenschaften Series, Vol. 173

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.

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.