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Z×Z denotes the Cartesian product of the set of integers with itself. It consists of all ordered pairs (m, n) where m and n are integers. Geometrically, Z×Z represents the integer lattice in the Euclidean plane: every lattice point has integer coordinates, forming a regular grid of unit squares.

Algebraically, Z×Z is an additive abelian group under componentwise addition: (m1, n1) + (m2, n2) = (m1 + m2, n1 + n2). It is a free abelian group of rank 2 and is isomorphic to Z^2. Every element can be uniquely expressed as m(1,0) + n(0,1), making (1,0) and (0,1) a standard basis for the lattice. The lattice is discrete in R^2 and has a fundamental region given by the unit square, which has area 1. The distance between two points (a, b) and (c, d) is the Euclidean distance sqrt((a − c)^2 + (b − d)^2).

Z×Z also carries a rich algebraic structure for higher mathematics. The automorphism group GL(2, Z) acts on Z×Z by change of basis, preserving the lattice. This finite generation and rigidity underpin various topics in number theory, geometry of numbers, and crystallography, where Z×Z serves as a basic two-dimensional lattice model. In broader contexts, Z×Z generalizes to Z^n, the standard lattice in higher-dimensional spaces, used in lattice-based cryptography and computational geometry.