Integer lattice Z^d

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

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

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

Algebraic structure. Addition and subtraction are defined componentwise:

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.