Integer lattice Z^d - Knowlpedia

Fix a positive integer $d$ . The integer lattice in dimension $d$ is

\mathbb{Z}^d=\{(x_1,\dots,x_d): x_i\in \mathbb{Z}\},

where $\mathbb{Z}$ denotes the integers.

Elements $x\in\mathbb{Z}^d$ are often called lattice sites or lattice points.

Remarks

Algebraic structure. Addition and subtraction are defined componentwise:

x+y=(x_1+y_1,\dots,x_d+y_d),\qquad x-y=(x_1-y_1,\dots,x_d-y_d).

For $a\in\mathbb{Z}^d$ , the map $x\mapsto x+a$ is a translation of the lattice.

Standard basis. For $i\in\{1,\dots,d\}$ , the vector $e_i\in\mathbb{Z}^d$ is the point with a 1 in the $i$ -th coordinate and 0 elsewhere. Many local notions on $\mathbb{Z}^d$ are described using these vectors.

A key combinatorial structure on $\mathbb{Z}^d$ is the nearest-neighbor-zd adjacency relation, which turns $\mathbb{Z}^d$ into an infinite graph.