Connections with Model Theory
Sizes of Subdirectly Irreducible Algebras
Bounding how large the subdirectly irreducible members of a variety can be, and the compactness arguments that produce the bounds.
Learning objectives
- State the size-bounding results
- Follow the compactness argument
- Connect bounds on irreducibles to structural conclusions
Why size matters
Birkhoff's theorem represents every algebra as a subdirect product of subdirectly irreducible algebras but gives no control over them. Bounding their size converts the representation from an existence statement into a classification tool.
The compactness argument
Suppose a variety V has definable principal congruences, so that subdirect irreducibility is expressible by a first-order sentence σ. If V has subdirectly irreducible members of unbounded finite size, then it has an infinite subdirectly irreducible member.
- Let Σ be the theory of V together with σ and, for each n, a sentence asserting the existence of at least n distinct elements.
- Every finite subset of Σ is satisfiable, witnessed by a sufficiently large finite subdirectly irreducible member.
- By compactness, Σ has a model — an infinite subdirectly irreducible algebra in V.
If a variety has no infinite subdirectly irreducible members, then its finite ones are bounded in size. This is the form applied in practice, and it converts a qualitative statement into a quantitative one.
Where the bounds come from
| Hypothesis | Bound obtained |
|---|---|
| Congruence-distributive, generated by a finite algebra A | All irreducibles lie in HS(A); size at most |A| |
| Discriminator variety generated by finite K | Irreducibles are the simple members of IS(K) |
| Definable principal congruences plus no infinite irreducibles | Finite bound by compactness |
| Semisimple with finitely many simple members | Bounded by the largest simple member |
| Arbitrary variety | No bound available |
For congruence-distributive varieties, Jónsson's lemma provides the bound directly and without compactness. The compactness argument is the tool for varieties where Jónsson does not apply or where the generating class is infinite.
Consequences of a bound
- Decidable equational theory. An identity holds throughout the variety exactly when it holds in each of the finitely many irreducibles, which is a finite check.
- Finite subvariety lattice. A subvariety is determined by which irreducibles it contains, so there are at most 2k subvarieties for k irreducibles.
- Residual smallness. A variety whose irreducibles have bounded size is called residually small; if all are finite of bounded size it is residually finite in a strong sense.
- Finite basis. Baker's theorem converts the bound into a finite equational basis for congruence-distributive varieties.
A variety is residually small if there is a cardinal bound on the size of its subdirectly irreducible members; residually finite if all are finite; residually less than n if all have fewer than n elements.
The residual character of a variety is one of its most informative invariants, and much of the post-1981 literature on the structure of varieties is organised around it.
Frequently asked questions
Can a finitely generated variety have infinite subdirectly irreducibles?
Yes, if it is not congruence-distributive. There are finitely generated varieties of groups with infinite subdirectly irreducible members.
Is residual smallness decidable?
For a finitely generated congruence-modular variety, McKenzie gave a characterisation; in general the question is difficult and connects to the decidability results of §5.
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.3, book pages 256-259.
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.