Connections with Model Theory
Filters, Reduced Products and the Construction
The reduced product construction in detail: how a filter on the index set determines which coordinates matter.
Learning objectives
- Construct the reduced product and verify it is well defined
- Interpret filters as notions of largeness
- Identify the degenerate cases
Filters as largeness
A filter on an index set I is a filter of the Boolean algebra Su(I) — a family of subsets closed upward and under finite intersection. The intuition is that members of the filter are the “large” sets.
| Filter | Large sets are | Notes |
|---|---|---|
| {I} | Only I itself | Trivial; reduced product is the direct product |
| Su(I) | Everything, including ∅ | Improper; the reduced product is trivial |
| Principal filter above J | Supersets of J | Reduced product is the product over J |
| Fréchet filter | Cofinite sets | Proper for infinite I |
| Ultrafilter | Exactly one of each set and its complement | The maximal case |
The construction
For a family (Ai) and filter F on I, define θF on ∏i Ai by a θF b if {i : a(i) = b(i)} ∈ F. The reduced product is ∏i Ai/F.
Reflexivity holds because the whole index set is in every filter; symmetry is immediate; transitivity uses closure under intersection; and the substitution property uses closure under intersection together with upward closure.
Degenerate cases
- Trivial filter {I}. Only identical tuples are identified, so the reduced product is the direct product itself.
- Improper filter. If ∅ belongs to F, all tuples are identified and the reduced product is a one-element algebra.
- Principal ultrafilter at i. The reduced product is isomorphic to Ai, so nothing new is constructed.
Every interesting application — the compactness theorem, non-standard models, Jónsson's lemma — requires a free ultrafilter, and hence the Boolean prime ideal theorem. Without choice principles the construction has no content.
What reduced products preserve
A reduced product of algebras satisfying an identity satisfies it, since the reduced product is a quotient of a product and both operations preserve identities.
More is true and less is true. Reduced products preserve exactly the Horn sentences — a class strictly between identities and all first-order sentences. Ultraproducts, being reduced products modulo ultrafilters, preserve all first-order sentences by Łoś's theorem.
| Construction | Preserves |
|---|---|
| Direct product | Horn sentences, including identities |
| Reduced product | Horn sentences |
| Ultraproduct | All first-order sentences |
| Substructure | Universal sentences |
| Homomorphic image | Positive sentences |
Chapter V §2 is organised around this table. Each row is a preservation theorem, and the reduced product row is the one that requires the most work — it is treated on the Horn sentence page.
Frequently asked questions
Can the reduced product be empty?
No. The universes are non-empty and the product of non-empty sets is non-empty by choice, so the quotient is non-empty.
Does the reduced product depend on the filter or only on the ultrafilters above it?
On the filter itself. Different filters give different congruences and generally non-isomorphic reduced products.
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 V.2, book pages 234-239.
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.