Complete lattice

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A conditionally complete lattice is one that satisfies at least one of these properties for bounded subsets. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete.

Complete lattices appear in many applications in mathematics and computer science. Both order theory and universal algebra study them as a special instance of lattices.

Complete lattices must not be confused with complete partial orders (cpos), a more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales).^{[citation needed]}