Terms, Free Algebras and Equational Logic

Mal'cev Conditions II: Congruence Distributivity and Jonsson Terms

Distributivity needs a chain of terms rather than one, and it buys Jónsson's lemma — the sharpest control over subdirectly irreducibles anywhere in the subject.

Engineering · Mathematics4 min readKV-MATH-0225

Learning objectives

State the Jónsson term condition and verify it for lattices.
Explain why distributivity is a weak rather than strong Mal'cev condition.
State Jónsson's lemma and its hypotheses.
Derive the bound on subdirectly irreducibles in a finitely generated CD variety.
Distinguish congruence-modular from congruence-distributive.
Identify arithmetical varieties as the intersection of the two conditions.

01Jónsson terms

Congruence-distributivity is characterised by the existence of a chain of ternary terms of unspecified length.

d₀(x,y,z) ≈ x dₙ(x,y,z) ≈ z
dᵢ(x,y,x) ≈ x for all i
dᵢ(x,x,z) ≈ d_i+1(x,x,z) for i even
dᵢ(x,z,z) ≈ d_i+1(x,z,z) for i odd

A variety is congruence-distributive iff such terms d₀,…,dₙ exist for some n. The length n is not bounded in advance, which makes this a weak Mal'cev condition.

For lattices, three terms suffice: d₀ = x, d₁(x,y,z) = (x ∧ y) ∨ (y ∧ z) ∨ (x ∧ z), d₂ = z. The middle term is the median, and verifying the identities is a short computation with the absorption laws.

02Weak versus strong

Permutability

Strong

One term, two identities, fixed shape. A single finite search decides it.

Distributivity

Weak

A chain of terms whose length is not bounded. Deciding it requires searching over increasing n, so it is semidecidable in general.

CautionUnbounded chain length has practical consequences

For a finite algebra one can search for Jónsson terms of length 3, 4, 5 and so on, but without a bound there is no termination guarantee from the characterisation alone. In practice bounds are known for particular settings, and tools such as UACalc use them; do not assume a naive search terminates.

03Jónsson's lemma

The reward for congruence-distributivity is exceptionally tight control over which algebras can be subdirectly irreducible in the generated variety.

Key resultJónsson's lemma

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

Compare the general situation, where subdirectly irreducibles of V(K) can only be located in HSP(K), which is the whole variety and therefore no information at all. Jónsson's lemma replaces P by P_U and moves it inside, which is an enormous strengthening.

04The finite case

ProcedureBounding subdirectly irreducibles in a finitely generated CD variety

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

input: finite set K of finite algebras, V(K) congruence-distributive
ultraproducts of a FINITE set of FINITE algebras are isomorphic to members of K
(an ultraproduct of finitely many finite algebras collapses)
so P_U(K) ⊆ I(K)
Jónsson's lemma gives: SI members of V(K) ⊆ HS(K)
HS(K) is a finite set of finite algebras, computable from K
therefore V(K) has finitely many subdirectly irreducibles, all bounded by max|A|

This is the engine behind Baker's finite basis theorem: a finite bound on subdirectly irreducibles is exactly what is needed to construct a finite equational basis. Caveat: congruence-distributivity is essential — the conclusion is false without it.

The consequence is that a finitely generated congruence-distributive variety is residually finite with a computable bound, has a decidable equational theory in many cases, and is finitely based by Baker's theorem. Very little else in universal algebra delivers so much from one hypothesis.

05Congruence modularity and Day terms

Modularity is weaker than either permutability or distributivity, and is likewise characterised by terms — Day terms, a chain of quaternary terms.

The three conditions compared
Condition	Terms	Type	Implied by
Permutable	one ternary Mal'cev term	strong	—
Distributive	chain of ternary Jónsson terms	weak	—
Modular	chain of quaternary Day terms	weak	permutable or distributive
Arithmetical	one ternary Pixley term	strong	permutable and distributive

Modularity is the weakest of the useful conditions and is exactly what the commutator theory requires. Hagemann and Herrmann extended Smith's commutator from the permutable to the modular setting, which is why the centre and solvability are available for a much wider class than groups.

06Arithmetical varieties

A variety is arithmetical when it is both congruence-permutable and congruence-distributive. Pixley showed this is a strong Mal'cev condition: a single ternary term suffices.

Pixley term: p(x,y,y) ≈ x, p(x,y,x) ≈ x, p(y,y,x) ≈ x

A term satisfying these three identities exists exactly when the variety is arithmetical. Compare the Mal'cev term, which drops the middle identity.

Example

Boolean algebras

The term (x ∧ z) ∨ (x ∧ y′) ∨ (y′ ∧ z) is a Pixley term. Boolean algebras are the archetypal arithmetical variety.

Example

Heyting algebras

Arithmetical, and the source of much of the interest in arithmeticity for logic.

Where it leads

Discriminator varieties

Every discriminator variety is arithmetical, and the discriminator term is a particularly well-behaved Pixley term. This is the entry point to the Boolean Constructions stream.

Frequently asked

Does congruence-distributivity imply congruence-permutability?

No. Lattices are congruence-distributive and not congruence-permutable. The two conditions are independent, and their conjunction — arithmeticity — is strictly stronger than either.

Why does Jónsson's lemma need ultraproducts?

Because in the infinite case a subdirectly irreducible member of V(K) can fail to embed in any single member of K, but must be approximable by them. The ultraproduct is the construction that captures 'approximable by members of K', and it is why this otherwise purely algebraic lemma requires model-theoretic machinery.

Is arithmeticity common?

Less common than modularity but strikingly well behaved where it occurs. Boolean algebras, Heyting algebras, and all discriminator varieties are arithmetical. Groups and rings are permutable but not distributive, so not arithmetical.

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.