Varieties, Free Algebras and Equational Logic
Mal'cev Conditions and Congruence Permutability
Conditions on a variety expressed by the existence of terms satisfying prescribed identities, and Mal'cev's theorem characterising congruence permutability by a single ternary term.
Learning objectives
- State Mal'cev's theorem and identify Mal'cev terms in examples
- Explain what a Mal'cev condition is in general
- Connect permutability to the isomorphism theorems
Mal'cev's theorem
A variety V is congruence-permutable — every pair of congruences on every member permutes — if and only if there is a ternary term p such that V satisfies both p(x, y, y) ≈ x and p(x, x, z) ≈ z.
A ternary term satisfying those two identities.
| Variety | Mal'cev term | Permutable? |
|---|---|---|
| Groups | p(x,y,z) = xy−1z | Yes |
| Rings | x − y + z | Yes |
| R-modules | x − y + z | Yes |
| Quasigroups | A term built from the division operations | Yes |
| Boolean algebras | Exists | Yes |
| Lattices | None exists | No |
| Semigroups | None exists | No |
| Semilattices | None exists | No |
Why the term forces permutability
Suppose p is a Mal'cev term and θ, φ are congruences with ⟨a, b⟩ ∈ θ ∘ φ — so there is c with a θ c and c φ b. Consider d = p(a, c, b).
So θ ∘ φ ⊆ φ ∘ θ, and by symmetry the two are equal.
What Mal'cev conditions are in general
A condition on a variety asserting the existence of terms satisfying a prescribed finite set of identities. A weak Mal'cev condition allows a countable disjunction of such requirements.
Mal'cev conditions convert a statement quantifying over all algebras in a variety and all their congruences into a finite syntactic requirement: does a term with these properties exist? This makes the conditions checkable in principle, preserved under interpretation between varieties, and comparable to one another.
| Property | Term requirement | Due to |
|---|---|---|
| Congruence-permutable | One ternary Mal'cev term | Mal'cev |
| Congruence-distributive | Jónsson terms — a finite chain of ternary terms | Jónsson |
| Congruence-modular | Day terms — a finite chain of quaternary terms | Day |
| Arithmetical | Permutable and distributive; a Pixley term | Pixley |
| Congruence-n-permutable | A chain of n − 1 terms | Hagemann–Mitschke |
Consequences of permutability
- The second isomorphism theorem holds unconditionally. Permutability supplies the closure of Bθ that the general statement lacks.
- Congruences are determined by one class when the variety also has a constant, which is why normal subgroups and ideals suffice for groups and rings.
- Joins are simple. θ ∨ φ = θ ∘ φ, so no transitive closure is needed.
- Congruence-permutable implies congruence-modular. The converse fails.
- Direct decompositions are better behaved, since factor congruences reduce to complemented congruences.
Frequently asked questions
Why do lattices have no Mal'cev term?
Because lattice congruences do not permute — the two-element chain has congruences whose relational products differ in the two orders. By Mal'cev's theorem the absence of permutability rules out any such term.
Are Mal'cev conditions preserved under taking subvarieties?
Yes. If a term exists in a variety and satisfies the required identities there, it satisfies them in any subvariety. So Mal'cev conditions pass downward.
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 II.12, book pages 85-87.
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.