Boolean Constructions and Discriminator Varieties
Jónsson's Lemma for Congruence-Distributive Varieties
The theorem bounding the subdirectly irreducible algebras of a congruence-distributive variety, and the structural consequences that follow from it.
Learning objectives
- State Jónsson's lemma precisely
- Derive the finitely generated case
- List the consequences for subvariety lattices and finite bases
The statement
Let K be a class of algebras such that V(K) is congruence-distributive. Then every subdirectly irreducible member of V(K) belongs to HSPU(K).
Without congruence-distributivity no comparable bound is available. The lemma is the principal reason congruence-distributivity is such a valuable hypothesis.
If K is a finite set of finite algebras, then PU(K) adds nothing — an ultraproduct of finitely many finite algebras is isomorphic to one of them. So the subdirectly irreducible members lie in HS(K), a finite set of finite algebras.
Consequences of the finitely generated case
| Consequence | Reason |
|---|---|
| Finitely many subdirectly irreducibles | All lie in HS(K), which is finite |
| All subdirect irreducibles are finite | Subalgebras and quotients of finite algebras are finite |
| Finitely many subvarieties | A subvariety is determined by which subdirect irreducibles it contains |
| Every member is a subdirect product of finitely many types | Birkhoff, with the bounded set of factors |
| Equational theory is decidable | Check identities in the finitely many finite irreducibles |
| A finite equational basis exists | Baker's theorem, developed in Chapter V §4 |
Worked instances
Boolean algebras
V({2}) is congruence-distributive and generated by a single two-element algebra. HS({2}) = {2, trivial}, so 2 is the only non-trivial subdirect irreducible — recovering the result obtained earlier by direct filter analysis.
Distributive lattices
Generated by the two-element chain, congruence-distributive. The same argument gives exactly one non-trivial subdirect irreducible, hence exactly two subvarieties above the trivial one, and a finite equational basis.
Lattices — where it does not help
The variety of all lattices is congruence-distributive but is not finitely generated. Jónsson's lemma applies but the bound HSPU(K) is uninformative when K must be infinite. Lattices have continuum many subvarieties.
Congruence-distributivity alone gives the lemma but not the strong consequences; finite generation alone gives nothing without distributivity. It is the conjunction that produces the finite structure theory.
The proof idea
The argument analyses a congruence on a subdirect product. Given a subdirectly irreducible A in V(K), write it as a quotient of a subalgebra of a product of members of K. The monolith of A pulls back to a congruence on that subalgebra.
Congruence-distributivity is used to show that the relevant congruence is determined by an ultrafilter on the index set — distributivity allows a join of congruences to be analysed componentwise, and the sets of coordinates involved form a filter which extends to an ultrafilter. Taking the corresponding ultraproduct gives the required membership in HSPU(K).
The componentwise analysis genuinely requires distributivity. In a congruence-modular variety the join of congruences does not decompose in the same way, and no analogue of Jónsson's lemma is available. This is the sharpest illustration of the gap between the two conditions.
Frequently asked questions
Does Jónsson's lemma have a converse?
Not as stated. The lemma gives an upper bound on the subdirect irreducibles; the corresponding lower bound — that everything in HSP_U(K) that is subdirectly irreducible actually occurs — requires separate argument and is not automatic.
Is there an analogue for congruence-modular varieties?
There are partial results using commutator theory, but nothing of comparable strength. Freese and McKenzie's work on congruence-modular varieties supplies the closest substitutes.
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 165-168.
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.