lattices

Lattices in mathematics refer to two related but distinct concepts. In order theory, a lattice is a partially ordered set in which any two elements have a least upper bound (join) and a greatest lower bound (meet). These operations satisfy the usual lattice laws and can give rise to finite or infinite structures, including special classes such as distributive, modular, or complemented lattices. In geometry, a lattice is a discrete subgroup of real n-space generated by integer linear combinations of n linearly independent vectors, also known as a lattice with a basis. Although the two notions share terminology, they belong to different areas of study.

Abstract lattice theory provides a framework for organizing elements by order and for studying algebraic properties of joins and meets. Examples include the power set of a set under union and intersection, or the lattice of divisors of an integer under gcd and lcm. The dual lattice concept is obtained by reversing the order.

Geometric lattices, or lattices in Euclidean space, are captured by a basis. The lattice consists of all integer combinations of the basis vectors; the rank equals the dimension of the space and the determinant (covolume) is the absolute value of the determinant of a basis matrix. Common examples are the integer lattice Z^n and various crystallographic lattices. Topics include lattice reduction, Voronoi cells, and density.

Applications of lattice theory appear in mathematics, computer science, cryptography, coding theory, and physics. In crystallography, lattices model the periodic arrangement of atoms; in cryptography, lattice-based schemes are studied for their security properties. In optimization and combinatorics, lattices provide tools for structuring data and solving problems with order constraints.