Boolean Constructions and Discriminator Varieties
Ultraproducts in Universal Algebra
The ultraproduct construction from the algebraic side: reduced products modulo an ultrafilter, and the properties they inherit.
Learning objectives
- Construct the ultraproduct of a family of algebras
- Show that varieties are closed under ultraproducts
- Distinguish the algebraic and model-theoretic roles of the construction
The construction
For a family (Ai)i∈I and a filter F on I, define θF on the product by a θF b if and only if {i : a(i) = b(i)} ∈ F. The reduced product is the quotient.
A reduced product modulo an ultrafilter U, written ∏i∈I Ai/U.
Two tuples are identified when they agree on a set that the filter regards as large. Since a filter is closed under finite intersection and upward inclusion, this is a congruence on the direct product.
- P<sub><em>R</em></sub>(<em>K</em>)
- reduced products of members of K
- P<sub><em>U</em></sub>(<em>K</em>)
- ultraproducts of members of K
- Ultrapower
- All factors equal: AI/U
- Principal <em>U</em>
- The ultraproduct is isomorphic to one factor
Closure properties
PU(V) ⊆ V for any variety V, since an ultraproduct is a quotient of a product.
More is true: reduced products and ultraproducts are both instances of HP, so they add nothing to a class already closed under H and P.
For classes not closed under H — quasivarieties, elementary classes, and the class of subdirectly irreducible members of a variety — ultraproducts are a genuinely new operation. Mal'cev's characterisation of quasivarieties uses closure under ISPPU, and Jónsson's lemma uses PU essentially.
Finiteness is not preserved
An ultraproduct of finite algebras need not be finite. If the sizes are unbounded and U is free, the ultraproduct is infinite. This is the mechanism behind non-standard models and behind many compactness arguments.
| Property | Preserved? |
|---|---|
| Satisfying a fixed identity | Yes |
| Satisfying a fixed first-order sentence | Yes — Łoś's theorem |
| Membership in a variety | Yes |
| Being finite | No |
| Having size exactly n | Yes — expressible by a sentence |
| Being simple | Not in general |
| Being subdirectly irreducible | Not in general |
The row that matters most is the second: ultraproducts preserve all first-order properties, which is Łoś's theorem, developed on the model-theoretic side in Chapter V.
The algebraic use
Within Chapter IV the ultraproduct is used for one purpose above all: Jónsson's lemma, which bounds the subdirectly irreducible members of a congruence-distributive variety by HSPU(K).
The full product operator would be far too weak a bound — every algebra in the variety lies in HSP(K) by definition. Restricting to ultraproducts is what makes the bound informative, and for finite K of finite algebras it collapses to HS(K).
Frequently asked questions
Is the ultraproduct of a family always non-trivial?
It is trivial only if the factors are trivial on a large set. Otherwise the ultraproduct has at least as many elements as a typical factor.
Why do reduced products matter separately from ultraproducts?
Because reduced products preserve Horn sentences, a strictly larger class than identities but smaller than all first-order sentences. Chapter V §2 develops that preservation theorem.
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 IV.6, book pages 163-165.
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.