Model-Theoretic Connections

Reduced Products and Filtered Products

A reduced product identifies elements agreeing on a large set, where a filter decides largeness. It sits between the direct product and the ultraproduct, and Horn sentences are exactly what survives.

Engineering · Mathematics5 min readKV-MATH-0250

Learning objectives

Construct the reduced product of a family over a filter.
Verify the induced relation is a congruence.
Identify direct products and ultraproducts as special cases.
State which sentences are preserved by reduced products.
Define Horn sentences and recognise them.
Explain why identities and quasi-identities are Horn.

01The construction

Given structures indexed by I and a filter F on the power set of I, identify two elements of the direct product when the set of coordinates where they agree lies in F.

θ_F := { ⟨f, g⟩ : ⟦f = g⟧ ∈ F } where ⟦f = g⟧ := { i : f(i) = g(i) }
∏_iA_i/F := ∏_iA_i / θ_F

The double-bracket notation for agreement sets matches the Boolean product usage in Chapter IV — deliberately, since the constructions are related.

ProcedureVerifying θ_F is a congruence

in: family A_i, filter F → out: θ_F is a congruence

reflexive: ⟦f = f⟧ = I ∈ F since F is a filter
symmetric: ⟦f = g⟧ = ⟦g = f⟧
transitive: ⟦f = g⟧ ∩ ⟦g = h⟧ ⊆ ⟦f = h⟧
F is closed under intersection and upward closed, so ⟦f = h⟧ ∈ F
compatible: if ⟦f_k = g_k⟧ ∈ F for k = 1,…,n then
⋂_k ⟦f_k = g_k⟧ ⊆ ⟦F(f⃗) = F(g⃗)⟧ for any operation F
finite intersection stays in F, so the image agrees on a large set
therefore θ_F ∈ Con(∏ A_i)

Every filter axiom is used exactly once: I ∈ F for reflexivity, intersection closure for transitivity and compatibility, upward closure to conclude membership. Caveat: relations are handled separately — r holds in the quotient when it holds on a set in F.

Relation symbols need a separate clause because the quotient must decide whether r(f₁,…,fₘ) holds. The convention is that it holds when the set of coordinates where it holds is in F. For algebras there are no relations and the question does not arise.

02The two extremes

Reduced products by choice of filter
Filter F	θ_F	Result
{I} — the trivial filter	Δ	the direct product itself
principal at i₀	agreement at i₀	isomorphic to A_{i₀}
the improper filter (all subsets)	∇	the trivial one-element structure
Fréchet filter (cofinite sets)	eventual agreement	a genuine reduced product
an ultrafilter U	θ_U	the ultraproduct

The ultraproduct is the case where the filter is maximal, and it is the only case where all first-order sentences transfer. For a general filter only Horn sentences do, which is the theorem below.

03Horn sentences

A basic Horn formula is a disjunction of literals with at most one of them positive. A Horn sentence is built from these by conjunction and universal quantification.

Horn forms and their readings
Form	Equivalent reading	Example
¬φ₁ ∨ ⋯ ∨ ¬φₙ ∨ ψ	(φ₁ & ⋯ & φₙ) → ψ	a quasi-identity
ψ	a positive atomic assertion	an identity
¬φ₁ ∨ ⋯ ∨ ¬φₙ	¬(φ₁ & ⋯ & φₙ)	a negative condition

Key resultHorn sentences are preserved by reduced products

If every A_i satisfies a Horn sentence σ, then so does every reduced product ∏A_i/F. This is the reduced product preservation theorem, and identities and quasi-identities are the cases that matter algebraically.

Identities are Horn: an identity p ≈ q is a single positive literal universally quantified. Quasi-identities are Horn: an implication between a conjunction of equations and one equation. So both classes transfer to reduced products, which is why varieties and quasivarieties are closed under the construction.

04Why the restriction to Horn

The proof of preservation uses the filter's closure under finite intersection, and it breaks precisely where a disjunction with two positive literals appears.

Positive atomic case
If ψ holds in every factor it holds on I ∈ F, so it holds in the quotient. No difficulty.
Implication case
If the hypotheses hold on a set in F, the conclusion holds there too by the factor hypothesis. Intersection closure keeps the set large.
Where it fails: two positive literals
A sentence ψ₁ ∨ ψ₂ might have ψ₁ holding on some coordinates and ψ₂ on others, with neither set in F. The disjunction then fails in the quotient though it holds in every factor.
Ultrafilters repair this
An ultrafilter decides every subset, so one of the two sets is in F. This is exactly why ultraproducts preserve all sentences and reduced products do not.

CautionThe failure is real, not an artefact of the proof

Explicit examples exist of non-Horn sentences holding in every factor and failing in a reduced product. The restriction to Horn sentences is a characterisation, not merely the limit of one proof technique.

05Filtered products in the Boolean setting

Reduced product

Filter on a set

Indexed by an arbitrary set with a filter on its power set. The general model-theoretic construction.

Filtered Boolean power

Filter data on the values

The Chapter IV construction. Related but distinct — the filtering constrains which locally constant functions are admitted, rather than which coordinates count as large.

The two notions are frequently confused because both are called 'filtered'. The reduced product filters the index set; the filtered Boolean power filters the value assignments. Both appear in the Burris and Werner decidability argument and are used for different purposes.

06The characterisation theorem

Preservation by reduced products characterises Horn sentences up to logical equivalence, giving a Birkhoff-style correspondence for this construction.

Key resultKeisler–Galvin characterisation

A first-order sentence is preserved under reduced products if and only if it is logically equivalent to a Horn sentence. Syntax and preservation coincide exactly, as in Birkhoff's theorem.

This is the pattern the whole preservation-theorem programme follows and which the next pages continue: identify a construction, identify the syntactic class preserved by it, and prove the two coincide. Birkhoff's HSP theorem is the equational instance, and each preservation theorem is a further instance.

Frequently asked

Is a reduced product over the Fréchet filter interesting?

Yes — it identifies sequences agreeing eventually, which is the natural notion for limit constructions. It is not an ultraproduct, since the Fréchet filter is not maximal, so only Horn sentences transfer. Extending it to an ultrafilter is what gives the full Łoś theorem.

Do reduced products preserve identities?

Yes, since identities are Horn. This is consistent with varieties being closed under P and H: a reduced product is a quotient of a direct product, so closure under P and H already guarantees it for varieties.

Why do relations need a separate clause?

Because the quotient must assign a truth value to r on equivalence classes, and the natural definition — r holds when it holds on a set in the filter — is a choice that must be made explicitly. For algebras with no relation symbols the question is vacuous, which is why the algebraic literature often omits the clause.

Sources and further reading

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.

Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.