Lattice Theory Foundations
Closure Operators and Algebraic Closure
Closure operators, the complete lattices of closed sets they generate, and the correspondence that makes them the organising device behind subuniverses, congruences and generated substructures alike.
Learning objectives
- State the three closure-operator axioms and verify them in examples
- Establish the correspondence between closure operators and closure systems
- Distinguish algebraic closure operators and connect them to algebraic lattices
Closure operators
A map C from Su(A) to Su(A) is a closure operator if for all X, Y ⊆ A: (i) X ⊆ C(X) — extensive; (ii) X ⊆ Y implies C(X) ⊆ C(Y) — monotone; (iii) C(C(X)) = C(X) — idempotent.
A set is closed if C(X) = X. The family of closed sets is written LC.
Closure operators and closure systems
A family of subsets of A containing A itself and closed under arbitrary intersection.
Closure operators on A and closure systems on A correspond bijectively:
- Given a closure operator C, the closed sets LC form a closure system.
- Given a closure system ℱ, the map X ↦ ⋂{Y ∈ ℱ : X ⊆ Y} is a closure operator.
- The two constructions are mutually inverse.
Algebraic closure operators
A closure operator C is algebraic if for every X and every a ∈ C(X) there is a finite Y ⊆ X with a ∈ C(Y).
The condition says membership in a closure is always witnessed by finitely much of the input. It is precisely what finitary operations deliver.
The lattice of closed sets of a closure operator is an algebraic lattice if and only if the operator is algebraic. In that case the compact elements are exactly the closures of finite sets.
| Operator | Closed sets | Algebraic? |
|---|---|---|
| Sg — subuniverse generated | Sub(A) | Yes |
| Θ — congruence generated | Con(A) | Yes |
| Subgroup generated | Subgroups | Yes |
| Ideal generated | Ideals | Yes |
| Deductive closure in logic | Theories | Yes — proofs are finite |
| Topological closure | Closed sets | No — not generally |
| Convex hull in the plane | Convex sets | Yes, by Carathéodory |
Topological closure is a closure operator but is not algebraic: a point can lie in the closure of a set without lying in the closure of any finite subset. This is exactly why the lattice of closed sets of a topological space need not be algebraic, and it isolates what finitary arity contributes in the algebraic setting.
Galois connections
Closure operators arise systematically from Galois connections. Given a relation between two sets, the two induced maps — each sending a subset to the set of everything related to all of it — compose to give closure operators on each side.
This particular Galois connection is the backbone of Chapter II §11 and §14: its closed classes of algebras are exactly the varieties, and its closed sets of identities are exactly the equational theories. Birkhoff's theorem and the completeness of equational logic are the two halves of describing it.
Frequently asked questions
Is every complete lattice the lattice of closed sets of a closure operator?
Yes. Given a complete lattice L, take the underlying set and define the closure of a subset to be the down-set of its supremum. The closed sets are then order-isomorphic to L.
What distinguishes a closure system from a topology?
A topology's closed sets are closed under finite union and arbitrary intersection. A closure system requires only arbitrary intersection. So every topology's closed sets form a closure system, but not conversely.
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.5, book pages 20-24.
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.