Boolean Constructions and Discriminator Varieties

Ultraproducts and Jonsson's Lemma

Jónsson's lemma is the sharpest tool in the subject. Under congruence distributivity it locates every subdirectly irreducible member of a generated variety inside HS of ultraproducts of the generators.

Engineering · Mathematics5 min readKV-MATH-0238

Learning objectives

Construct the ultraproduct of a family of algebras.
State Jónsson's lemma with its hypotheses.
Explain why congruence distributivity is essential.
Derive the finite case where ultraproducts collapse.
List the structural consequences for finitely generated CD varieties.
Recognise where the lemma fails without distributivity.

01The ultraproduct construction

Given a family of algebras indexed by I and an ultrafilter U on I, the ultraproduct is the direct product modulo the congruence identifying elements that agree on a set in U.

θ_U := { ⟨f, g⟩ : { i ∈ I : f(i) = g(i) } ∈ U } ∏_iA_i/U := ∏_iA_i / θ_U

Two elements are identified when they agree on a 'large' set, where U decides largeness. θ_U is a congruence because U is a filter.

Over a principal ultrafilter the construction collapses to a single factor and gives nothing. Over a free ultrafilter it produces genuinely new algebras, and it is the mechanism behind compactness, non-standard models and — here — Jónsson's lemma. The detailed model theory, including Łoś's theorem, belongs to the Model-Theoretic stream.

02The statement

Key resultJónsson's lemma

If V(K) is congruence-distributive, then every subdirectly irreducible algebra in V(K) belongs to HS(P_U(K)) — the homomorphic images of subalgebras of ultraproducts of members of K.

Without the hypothesis

SI members lie in HSP(K)

Which is the whole variety. Completely uninformative — the subdirect representation theorem already tells us the subdirectly irreducibles are somewhere in there.

With congruence distributivity

SI members lie in HS(P_U(K))

The unrestricted product operator P is replaced by the far weaker P_U and moved inside. This is an enormous strengthening.

The gain comes from two places at once. Ultraproducts are much more restrictive than arbitrary products, and the reordering puts the product operator innermost, so no products of large families are formed after taking subalgebras and quotients.

03Why distributivity is essential

The proof works by analysing a congruence on a subdirect product using the distributive law in the congruence lattice.

Set up
A subdirectly irreducible A ∈ V(K) is a quotient of a subalgebra of a product of members of K. Let the projection kernels be θ_i.
The monolith forces concentration
Because A is subdirectly irreducible its monolith is the least non-trivial congruence. Pulling back, the corresponding congruence must not be split across the factors.
Distributivity does the splitting
In a distributive congruence lattice, a congruence meeting a finite join must meet one of the joinands. Iterating gives a family of indices that is closed under supersets and finite intersections — a filter.
The filter is an ultrafilter
Maximality of the analysis forces the filter to be an ultrafilter, and the corresponding quotient is an ultraproduct.

CautionThe conclusion is false without distributivity

For congruence-modular but not distributive varieties there is no analogue. Groups form a congruence-permutable, hence modular, variety, and the subdirectly irreducible groups in a variety generated by a finite group are not confined to HS of ultraproducts of it. Any attempt to apply the lemma outside the distributive setting is simply invalid.

04The finite case

ProcedureFinitely generated congruence-distributive varieties

in: finite K generating a CD variety → out: complete finite list of SIs

input: finite set K of finite algebras, V(K) congruence-distributive
an ultraproduct of a FINITE family of FINITE algebras is isomorphic to a factor
(the ultrafilter is principal on a finite index set, or concentrates)
more precisely: P_U(K) ⊆ I(K) when K is a finite set of finite algebras
Jónsson's lemma then gives: SI members of V(K) ⊆ HS(K)
HS(K) is a finite, explicitly computable set of finite algebras
bound: every SI member has at most max{ |A| : A ∈ K } elements

Correctness: the collapse of ultraproducts over finite index sets is what removes P_U entirely. Caveat: this needs K finite AND each member finite; either failing reintroduces genuine ultraproducts.

A finite bound on the subdirectly irreducibles is an extremely strong conclusion. It makes the variety residually finite with an explicit bound, and it is the hypothesis Baker's finite basis theorem needs.

05Consequences

Finite basis

Baker's theorem

A finite algebra generating a congruence-distributive variety is finitely based. The bounded subdirectly irreducibles are what make the equational basis constructible.

Residual finiteness

Bounded SIs

Every member is a subdirect product of algebras from a fixed finite list, so the variety is residually finite with an explicit bound.

Decidability

In favourable cases

Combined with the discriminator theory, gives decidable first-order theories for finitely generated discriminator varieties of finite type.

Lattice of subvarieties

Finitely many

A finitely generated CD variety has only finitely many subvarieties, since each is determined by which of the finitely many SIs it contains.

Structure

Complete description

Knowing all the subdirectly irreducibles plus Birkhoff's representation theorem determines the variety entirely.

Where it applies

Lattices, Boolean, Heyting

All congruence-distributive. The lemma applies to any variety of lattices with additional operations, which covers most algebras of logic.

06Scope and limits

Where Jónsson's lemma applies
Variety	CD?	Lemma applies?
Lattices and expansions	Yes	Yes
Boolean algebras	Yes	Yes, trivially — one SI
Heyting algebras	Yes	Yes
Discriminator varieties	Yes	Yes, and sharply
Groups	No	No — modular but not distributive
Rings	No	No
Modules	No	No
Quasigroups	No	No
Semigroups	No	No

The division is stark: the algebras of logic are congruence-distributive and the algebras of classical algebra are not. This is the single largest reason the structure theory of Chapter IV concerns lattice-like varieties rather than group-like ones, and why the commutator theory had to be developed separately for the modular case.

Frequently asked

Does Jónsson's lemma need the axiom of choice?

It needs ultrafilters, hence BPI. For finitely generated varieties over finite algebras the ultraproducts collapse and no choice is required, which is why the finite case is the one used computationally.

Is the converse of Jónsson's lemma true?

No. Containment of the subdirectly irreducibles in HS(P_U(K)) does not force congruence distributivity. The lemma is a one-way implication, and distributivity is a hypothesis rather than a characterisation.

What replaces Jónsson's lemma for modular varieties?

Nothing as sharp. The commutator theory of Smith, Hagemann–Herrmann and Freese–McKenzie provides tools for congruence-modular varieties, and finite basis results were later obtained in that setting, but there is no statement locating the subdirectly irreducibles as tightly. The gap between the modular and distributive cases is real.

Sources and further reading

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.

Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.