Lattice Theory Foundations
Complete Lattices and Completeness Criteria
Lattices in which every subset — not merely every pair — has a supremum and an infimum, and the surprisingly economical criterion that establishes completeness from one half of the condition alone.
Learning objectives
- Define complete lattice and identify complete examples
- Apply the one-sided criterion for completeness
- Explain why completeness matters for Sub(A) and Con(A)
Definition
A poset is a complete lattice if every subset — including the empty set — has both a least upper bound and a greatest lower bound. Joins and meets of arbitrary families are written ⋁ and ⋀.
Taking the empty set forces the existence of a greatest element 1 (its infimum) and a least element 0 (its supremum). Every complete lattice is therefore bounded.
In a finite lattice, every subset is finite, and finite joins and meets are obtained by iterating the binary operations. Completeness is a condition with content only in the infinite case.
The one-sided criterion
If a poset P has a greatest element and every non-empty subset has a greatest lower bound, then P is a complete lattice.
The proof constructs the missing suprema out of the given infima. Given A ⊆ P, let U be the set of upper bounds of A. This set is non-empty, since the greatest element belongs to it, so inf U exists by hypothesis. That infimum is an upper bound of A and is below every other one, so it is sup A.
Verifying completeness directly means checking two conditions for every subset. The criterion halves the work — and in the cases that matter, the infima are the easy half. For subuniverses and congruences, arbitrary intersections are again subuniverses and congruences, giving all infima for free; the criterion then delivers completeness without computing a single join.
The examples that matter
| Lattice | Infimum | Supremum | Complete? |
|---|---|---|---|
| Su(A), the power set | Intersection | Union | Yes |
| Sub(A), subuniverses | Intersection | Subuniverse generated by the union | Yes |
| Con(A), congruences | Intersection | Congruence generated by the union | Yes |
| Eq(A), equivalence relations | Intersection | Transitive closure of the union | Yes |
| Closed sets of a closure operator | Intersection | Closure of the union | Yes |
| The rationals under ≤ | — | — | No |
| Finite subsets of an infinite set | — | — | No |
In Sub(A), Con(A) and Eq(A) the join is not the set-theoretic union — the union of two subuniverses is rarely a subuniverse, and the union of two equivalence relations is rarely transitive. The join is the closure of the union. This asymmetry between meet and join is responsible for much of the difficulty in computing congruence lattices.
Completeness and closure systems
Every complete lattice arises as the lattice of closed sets of a closure operator on some set, and conversely the closed sets of any closure operator form a complete lattice under inclusion. This correspondence is developed in detail on the closure-operators page and is the mechanism by which Sub(A) and Con(A) are shown complete.
Frequently asked questions
Does every lattice embed in a complete lattice?
Yes. The Dedekind–MacNeille completion embeds any poset into a complete lattice preserving all existing suprema and infima, and it is the smallest such completion.
Why include the empty set in the definition?
Because it forces top and bottom elements to exist, which simplifies every subsequent statement. Excluding it gives a slightly weaker notion that would require carrying boundedness as a separate hypothesis.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section I.4, book pages 17-18.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.