Connections with Model Theory
Theories, Models and Axiomatisability
Theories as sets of sentences, model classes, and the question of which classes of algebras are first-order axiomatisable.
Learning objectives
- Define theory, model class and elementary class
- State the characterisation of elementary classes
- Locate varieties within the axiomatisability hierarchy
Theories and models
- Th(<strong>A</strong>)
- the set of all sentences true in A
- Th(<em>K</em>)
- the sentences true in every member of K
- Mod(Σ)
- the class of all models of a set of sentences Σ
- Th<sub>∀</sub>(<em>K</em>)
- the universal sentences true throughout K
- Complete theory
- one deciding every sentence: Φ or ¬Φ belongs
Th and Mod form a Galois connection exactly parallel to Id and M from Chapter II, but over the full first-order language rather than identities alone.
A class of the form Mod(Σ) for some set Σ of first-order sentences. A basic elementary class is one where Σ may be taken finite.
Characterising elementary classes
A class K closed under isomorphism is elementary if and only if both K and its complement are closed under ultraproducts — equivalently, K is closed under ultraproducts and elementary equivalence.
The condition on the complement is what distinguishes elementary classes from classes merely closed under ultraproducts.
| Class | Elementary? | Reason |
|---|---|---|
| Groups | Yes — basic | Finitely many axioms |
| Fields | Yes — basic | Finitely many axioms |
| Algebraically closed fields | Yes, not basic | One axiom per degree; infinitely many needed |
| Torsion-free abelian groups | Yes, not basic | One axiom per n |
| Torsion abelian groups | No | Not closed under ultraproducts |
| Finite groups | No | Ultraproducts can be infinite |
| Simple groups | No | Not closed under ultraproducts |
| Well-ordered sets | No | Compactness produces infinite descending chains |
To show a class is not elementary, produce members whose ultraproduct escapes the class. Finite groups of unbounded order have infinite ultraproducts; torsion groups of unbounded exponent have ultraproducts with elements of infinite order. Both arguments are two lines once the machinery is in place.
Where varieties sit
| Class type | Axioms | Closure properties |
|---|---|---|
| Variety | Identities | H, S, P |
| Quasivariety | Quasi-identities | I, S, P, PU |
| Universal class | Universal sentences | I, S, PU |
| Elementary class | Arbitrary sentences | I, PU, elementary equivalence |
Every variety is an elementary class, since identities are sentences. The converse fails badly: fields form an elementary class and are not a variety.
Whether a variety is a basic elementary class is exactly the finite basis question for its equational theory. Chapter V §4 addresses when a finite basis exists, and it is the point where the model-theoretic and equational threads meet.
Completeness of theories
A theory is complete when it decides every sentence, equivalently when all its models are elementarily equivalent.
| Theory | Complete? |
|---|---|
| Th(A) for any single structure | Yes, by construction |
| Theory of groups | No — abelian and non-abelian groups both model it |
| Theory of algebraically closed fields of characteristic 0 | Yes |
| Theory of dense linear orders without endpoints | Yes |
| Theory of atomless Boolean algebras | Yes |
| Theory of Boolean algebras | No — finite and infinite ones differ |
A complete theory decides every sentence semantically, but there may be no algorithm computing which way. A theory is decidable when it is complete and the set of consequences is computably enumerable in a usable form. The distinction matters in §5.
Frequently asked questions
Is Th(K) always a complete theory?
Only when all members of K are elementarily equivalent. For a variety with diverse members, Th(K) is far from complete.
Why are finite structures problematic for elementary classes?
Because the class of finite structures of a given type is never elementary — compactness always produces an infinite model from structures of unbounded size.
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.1, book pages 230-234.
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.