Definition:Lattice (Group Theory)

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Definition

Definition 1

A (point) lattice is a discrete subgroup of $\R^m$ under addition.


Definition 2

Let $\R^m$ be the $m$-dimensional real Euclidean space.

Let $\set {b_1, b_2, \ldots, b_n}$ be a set of linearly independent vectors of $\R^m$.

A lattice in $\R^m$ is the set of all integer linear combinations of such vectors.


That is:

$\displaystyle \map \LL {b_1, b_2, \ldots, b_n} = \set {\sum_{i \mathop = 1}^n x_i b_i : x_i \in \Z}$