Connections with Model Theory
Horn Sentences and Reduced-Product Preservation
The syntactic class matching closure under reduced products, and why direct products behave so well for the classes algebra cares about.
Learning objectives
- Define Horn sentences and identify examples
- State the preservation theorem for reduced products
- Explain why identities and quasi-identities are Horn
Horn sentences
A disjunction of literals containing at most one positive literal — equivalently, an implication whose hypothesis is a conjunction of atomic formulas and whose conclusion is a single atomic formula or falsity.
A sentence built from basic Horn formulas using conjunction and universal quantification, in prenex form with a Horn matrix.
| Sentence | Horn? |
|---|---|
| ∀x (p ≈ q) — an identity | Yes |
| ∀x (∧pi ≈ qi → p ≈ q) — a quasi-identity | Yes |
| Cancellation: xz ≈ yz → x ≈ y | Yes |
| ∀x (x ≈ 0 ∨ ∃y xy ≈ 1) — field inverses | No — two positive literals |
| ∀x∀y (x ≤ y ∨ y ≤ x) — totality | No |
The field axiom stating that every non-zero element has an inverse is a disjunction with two positive parts. That is exactly the non-Horn shape, and it is exactly why fields are not closed under products — the preservation theorem explains the failure rather than merely recording it.
The preservation theorem
A sentence preserved under reduced products is logically equivalent to a Horn sentence, and conversely every Horn sentence is preserved under reduced products — in particular under direct products.
The forward direction is the substantial one and is due to work of Chang, Łoś and Horn. The converse is an induction on formula structure using filter closure properties.
Why the algebraic classes are Horn
Identities and quasi-identities are Horn, which explains a great deal.
| Class | Axioms | Horn? | Closed under products? |
|---|---|---|---|
| Varieties | Identities | Yes | Yes |
| Quasivarieties | Quasi-identities | Yes | Yes |
| Universal classes | Universal sentences | Not necessarily | Not necessarily |
| Elementary classes | Arbitrary | Not necessarily | Not necessarily |
Chapter II observed that varieties are closed under products and proved it directly. The Horn preservation theorem explains why: identities are Horn, and Horn sentences are exactly what products preserve. The algebraic fact is a special case of a logical one.
Mal'cev's theorem in this light
A class closed under isomorphism is a quasivariety if and only if it is closed under I, S, P and PU.
Compare with Birkhoff: dropping H and adding PU takes one from varieties to quasivarieties, and from identities to quasi-identities. The two theorems are the same result at adjacent levels of the syntactic hierarchy.
| Birkhoff | Mal'cev | |
|---|---|---|
| Axioms | Identities | Quasi-identities |
| Closure | H, S, P | I, S, P, PU |
| Class name | Variety | Quasivariety |
| Free algebras exist | Yes | Yes |
| Closed under quotients | Yes | No |
Varieties are closed under H, and PU is contained in HP — so ultraproduct closure is automatic. Without H it must be assumed separately.
Frequently asked questions
Are all sentences preserved by direct products Horn?
Up to logical equivalence, yes for reduced products. For direct products alone the situation is slightly more generous, but Horn is the clean characterisation and the one the source uses.
Why is cancellation a quasi-identity rather than an identity?
Because it is conditional — it asserts an equation only under a hypothesis. That conditional shape is precisely what makes cancellative semigroups closed under products and substructures but not under quotients.
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 249-252.
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.