1. Modular lattices arise naturally in algebra and in many other areas of mathematics.2. In a modular lattice , however, equality holds. 3. For instance, around 1900, he wrote the first papers on modular lattices . 4. Neumann generalized this to continuous geometries, and more generally to complemented modular lattices , as follows. 5. This result is sometimes called the "'diamond isomorphism theorem "'for modular lattices . 6. This example also shows that the lattice of all subgroups of a group is not a modular lattice in general. 7. Any complemented modular lattice having a " basis " of pairwise perspective elements, is isomorphic with the lattice of all principal regular ring. 8. Since a lattice is modular if and only if all pairs of elements are modular, clearly every modular lattice is M-symmetric. 9. However, the completion of a distributive lattice need not itself be distributive, and the completion of a modular lattice may not remain modular. 10. Her constructions include extremal even unimodular lattices in 48, 56, and 72 dimensions and an extremal 3-modular lattice in 64 dimensions.
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