complete lattice

A complete lattice strengthens the lattice idea from pairs to arbitrary collections: not only does every two elements have a meet and a join, but every subset — even an infinite one — has a greatest lower bound and a least upper bound. This completeness is what lets you take suprema and infima freely, which is the technical backbone behind fixed-point theorems and the construction of closures throughout mathematics.

Formally, a poset (L, ≤) is a complete lattice if every subset S of L has both an infimum (meet) ⋀S and a supremum (join) ⋁S in L. Taking S empty and S = L shows a complete lattice automatically has a least element 0 = ⋁∅ = ⋀L and a greatest element 1 = ⋀∅ = ⋁L. A useful economy: it suffices that all subsets have meets (or dually all have joins), since joins can be reconstructed from meets via ⋁S = ⋀{upper bounds of S}.

Complete lattices are ubiquitous as ‘lattices of substructures’: the subgroups of a group, the subspaces of a vector space, the ideals of a ring, the congruences of any algebra, the closed sets of a topological space, and the closure operators on a set all form complete lattices. The Knaster-Tarski theorem makes them indispensable in logic and computer science: every monotone map on a complete lattice has a complete lattice of fixed points, hence a least and a greatest fixed point — the foundation of inductive and coinductive definitions.