Lattice (group)

In geometry and group theory, a lattice in the real coordinate space

R

n

{\displaystyle \mathbb {R} ^{n}}

is an infinite set of lattice points having the following properties:

Coordinate-wise addition or subtraction of two points in the lattice produces another lattice point.

The lattice points are all separated by some minimum distance (isolated).

Every point in the space is within some maximum distance of a lattice point.

One of the simplest examples of a lattice is the square lattice, which consists of all points

(

a

,

b

)

{\displaystyle (a,b)}

in the plane whose coordinates are both integers, and its higher-dimensional analogues, the integer lattices

Z

n

{\displaystyle \mathbb {Z} ^{n}}

.

Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space. The requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set.

More abstractly, a lattice can be described as a free abelian group of dimension

n

{\displaystyle n}

which spans the vector space ⁠

R

n

{\displaystyle \mathbb {R} ^{n}}

⁠. For any basis of ⁠

R

n

{\displaystyle \mathbb {R} ^{n}}

⁠, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regular tiling of a space by a primitive cell.

Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory.