Recent Developments and Resources
The Classification of Varieties and Tame Congruence Theory
The programme of classifying locally finite varieties by the local structure of their finite members, developed after the source text.
Learning objectives
- Describe the aim of tame congruence theory
- List the five types and what each means
- Explain how type-omitting conditions classify varieties
Material on this page extends past the 1981 text and its Millennium re-typesetting. Statements here are attributed to later literature, not to Burris and Sankappanavar. Where the status of a question is unsettled, this page says so rather than resolving it.
The programme
Tame congruence theory analyses a finite algebra by examining its behaviour on small subsets — the minimal sets associated with covering pairs in the congruence lattice. Each such pair receives a type describing the local structure.
The set of types occurring among covering pairs in congruence lattices of finite members of the variety.
Tame congruence theory is due to Hobby and McKenzie, published as The Structure of Finite Algebras (1988). It postdates the source entirely. The source's Recent Developments chapter identifies the classification of varieties as an active direction; this is what that direction became.
The five types
| Type | Local structure | Model |
|---|---|---|
| 1 — unary | A set with permutations | G-sets |
| 2 — affine | A vector space or module | Modules |
| 3 — Boolean | The two-element Boolean algebra | Boolean algebras |
| 4 — lattice | The two-element lattice | Distributive lattices |
| 5 — semilattice | The two-element semilattice | Semilattices |
Each covering pair in a congruence lattice localises to one of these five behaviours. The type set of a variety records which occur.
Many Mal'cev conditions correspond exactly to omitting types. A locally finite variety is congruence-distributive if and only if it omits types 1, 2 and 5; congruence-modular if and only if it omits types 1 and 5; congruence-permutable if and only if it omits types 1, 4 and 5.
What the classification explains
| Variety | Types occurring |
|---|---|
| Boolean algebras | 3 |
| Distributive lattices | 3, 4 |
| Lattices | 3, 4 |
| K-vector spaces | 2 |
| Abelian groups | 2 |
| Groups | 1, 2, 3 |
| Semilattices | 5 |
| Semigroups | All five |
The table explains structural facts recorded earlier in the collection. Boolean algebras omit everything but type 3, which is why they are so rigid; semigroups admit all five types, which is why they resist structure theory entirely.
Consequences
- Decidability results. McKenzie and Valeriote characterised the decidable locally finite varieties in terms of omitted types.
- Constraint satisfaction. The algebraic CSP dichotomy is stated in terms of the presence or absence of certain term operations, and tame congruence theory supplies the underlying analysis.
- Finite basis theorems. Willard's theorem uses congruence-meet-semidistributivity, which is a type-omitting condition.
- A uniform explanation. The Mal'cev conditions of Chapter II §12, discovered independently, turn out to be instances of a single classification.
Tame congruence theory analyses finite algebras and applies to locally finite varieties. Varieties with infinite finitely generated members — groups, lattices in general — are outside its direct scope, though many results extend.
Frequently asked questions
Why exactly five types?
Because the local structure of a minimal set in a finite algebra is heavily constrained, and the classification of the possible induced algebras yields exactly these five. It is a theorem, not a definition.
Does the theory apply to a single finite algebra or to a variety?
Both. Types are assigned to covering pairs in a single finite algebra; the type set of a variety collects what occurs across all its finite members.
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 RD.2, book pages 284-285.
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.