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lattice in $\mathbb{R}^n$ (Definition)
Definition 1   A lattice in $ {\mathbb{R}}^n$ is an $ n$-dimensional additive free group over $ \mathbb{Z}$ which generates $ {\mathbb{R}}^n$ over $ \mathbb{R}$.

Example: The following is an example of a lattice $ \mathcal{L}\subset {\mathbb{R}}^2$, generated by $ w_1=(1,2),w_2=(4,1)$.

$\displaystyle \mathcal{L}=\{ \alpha w_1 +\beta w_2 \mid \alpha,\beta \in \mathbb{Z}\} $
    
\includegraphics[scale=0.75]{lattice}



"lattice in $\mathbb{R}^n$" is owned by alozano.
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See Also: Minkowski's theorem, Pick's theorem, product of posets

Other names:  lattice, grid
Also defines:  lattice in $\mathbb{R}^n$
Keywords:  lattice, grid
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Cross-references: generated by, generates, free group, additive
There are 50 references to this entry.

This is version 4 of lattice in $\mathbb{R}^n$, born on 2003-08-18, modified 2003-08-28.
Object id is 4613, canonical name is LatticeInMathbbRn.
Accessed 9218 times total.

Classification:
AMS MSC: 11H06 (Number theory :: Geometry of numbers :: Lattices and convex bodies)
Pending Errata and Addenda
None.
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Puzzling by alozano on 2005-03-02 10:19:21
I had not followed the thread about \mathbb{C} but this entry (lattice in \mathbb{R}^n) might be having some related trouble. Most puzzling of all, if you look at the bottom of the entry (at least in page images mode) it says "lattice in \sin(x)" is owned by alozano and also in the "also defines: lattice in \sin(x)". However, I never wrote \sin(x) in there and if one sees the entry, it says \mathbb{R}...

What on earth!?

Alvaro
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