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