Core Structure Theory
Subuniverses and the Generation Operator Sg
The operator Sg that produces the smallest subuniverse containing a given set, its two equivalent descriptions, and the finitary character that makes it an algebraic closure operator.
Learning objectives
- Define Sg from above and from below and prove the descriptions agree
- Compute Sg(X) in concrete algebras
- Establish that Sg is an algebraic closure operator
Two descriptions
For X ⊆ A, Sg(X) is the intersection of all subuniverses of A containing X.
This is well defined because A itself is such a subuniverse and intersections of subuniverses are subuniverses. It gives the smallest subuniverse containing X but says nothing about its elements.
Define E(X) to be X together with all elements fA(a1,…,an) where f is a basic operation and the arguments lie in X. Then iterate: X ⊆ E(X) ⊆ E2(X) ⊆ …
Sg(X) = X ∪ E(X) ∪ E2(X) ∪ … — the union of the finite iterates.
Why the iteration stops at ω
The union of the finite iterates is closed under every basic operation. Given arguments in the union, each lies in some Ek(X); since an operation takes only finitely many arguments, one may take the largest such k and conclude the result lies in Ek+1(X).
This argument is the reason Sg is algebraic: every element of Sg(X) already lies in Sg(Y) for some finite Y ⊆ X, because it was built by finitely many operations from finitely many generators. With infinitary operations the iteration would run past ω and the conclusion would fail.
Sg is extensive, monotone and idempotent, and for every X and every a ∈ Sg(X) there is a finite Y ⊆ X with a ∈ Sg(Y).
Computing Sg
| Algebra | Sg(<em>X</em>) equals |
|---|---|
| Group | The subgroup generated by X — all finite products of elements of X and their inverses |
| Ring with unit | All finite sums of finite products of elements of X, together with integer multiples of 1 |
| R-module | All finite R-linear combinations of X |
| Lattice | All elements obtained by finitely many joins and meets from X |
| Semigroup | All finite non-empty products of elements of X |
In every case Sg(X) is the set of values pA(a1,…,an) as p ranges over all terms and the arguments over X. That uniform description is proved once the term algebra is available, in §10, and it subsumes every row of the table.
Consequences
- Sub(A) is an algebraic lattice, with the finitely generated subuniverses as its compact elements.
- Directed unions are subuniverses. If a family of subuniverses is upward directed, its union is a subuniverse — a consequence of finitary arity used constantly in Zorn's lemma arguments.
- An algebra is generated by X when Sg(X) = A; a homomorphism is then determined by its values on X.
- Finitely generated algebras are those with Sg(X) = A for some finite X, and they play the role of compact objects throughout.
If Sg(X) = A and two homomorphisms from A agree on X, they agree everywhere. This is the fact that makes free algebras useful: specifying a homomorphism out of a free algebra amounts to specifying an arbitrary function on the free generators.
Frequently asked questions
Does Sg(∅) always make sense?
It equals the set of values of the constants in the type, closed under the operations. If the type has no constants, Sg(∅) is empty — a subuniverse but not a subalgebra.
Is E(X) itself a subuniverse?
Not in general — applying an operation to elements newly produced by E can escape E(X). That is exactly why the iteration is needed rather than a single step.
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 II.2-3, book pages 31-34.
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.