Lattice theory

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. The class of lattices can be generalized to semilattices, and some notable subclasses of lattices are Heyting algebras, Boolean algebras, distributive lattices, and geometric lattices (matroids). These lattice-like structures all admit order-theoretic as well as algebraic descriptions. The sub-field that studies lattices is called lattice theory.