Editing Complete lattice (section)

{{Short description|Partially ordered set in which all subsets have both a supremum and infimum}}
[[File:Hasse Diagram of Subgroup Lattice of D4.png|thumb|400px|The complete [[Lattice of subgroups|subgroup lattice]] for D4, the [[dihedral group]] of the square. This is an example of a complete lattice.]]

In [[mathematics]], a '''complete lattice''' is a [[partially ordered set]] in which all subsets have both a [[supremum]] ([[Join (mathematics)|join]]) and an [[infimum]] ([[Meet (Mathematics)|meet]]). A '''conditionally complete lattice''' satisfies at least one of these properties for bounded subsets. For comparison, in a general [[Lattice (order)|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 class of lattices.

Complete lattices must not be confused with [[complete partial order]]s (CPOs), a more general class of partially ordered sets. More specific complete lattices are [[complete Boolean algebra]]s and [[complete Heyting algebra]]s (locales).{{Citation needed|date=June 2024}}