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Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])
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Ordered lattice point enumeration
I am going to try to answer the version of the Observation which seems to imply that the original "orthogonal" basis condition really is meant to mean "positive multiples of the standard basis vectors …