Varieties, Free Algebras and Equational Logic
Congruence-Distributive and Congruence-Modular Varieties
Varieties in which every congruence lattice is distributive or modular, the term conditions characterising them, and the structure theory each unlocks.
Learning objectives
- Define congruence-distributive and congruence-modular varieties
- State the Jónsson and Day term characterisations
- Compare what each condition delivers structurally
The two conditions
A variety in which Con A is a distributive lattice for every member A.
A variety in which Con A is modular for every member A.
| Variety | Permutable | Distributive | Modular |
|---|---|---|---|
| Groups | Yes | No | Yes |
| Rings | Yes | No | Yes |
| R-modules | Yes | No | Yes |
| Lattices | No | Yes | Yes |
| Distributive lattices | No | Yes | Yes |
| Boolean algebras | Yes | Yes | Yes |
| Heyting algebras | No | Yes | Yes |
| Semigroups | No | No | No |
| Semilattices | No | No | No |
A variety both congruence-permutable and congruence-distributive is called arithmetical. Boolean algebras and all discriminator varieties are arithmetical, and Pixley's theorem characterises the condition by a single ternary term.
Term characterisations
A variety is congruence-distributive if and only if for some n there are ternary terms t0,…,tn satisfying t0(x,y,z) ≈ x, tn(x,y,z) ≈ z, ti(x,y,x) ≈ x for all i, and the alternating conditions ti(x,x,z) ≈ ti+1(x,x,z) for even i and ti(x,z,z) ≈ ti+1(x,z,z) for odd i.
A variety is congruence-modular if and only if there is a finite chain of quaternary Day terms satisfying an analogous system of identities.
Both characterisations replace a lattice condition on all congruences of all members by the existence of finitely many terms. The number of terms needed is a genuine invariant — lattices need three Jónsson terms; some varieties need many more.
What congruence-distributivity delivers
Distributivity is by some margin the stronger and more useful of the two.
None of these has an analogue for congruence-modular varieties in general. The gap between the two conditions is where much of the difficulty of modern universal algebra lies.
What congruence-modularity delivers
Modularity is weaker but covers groups, rings and modules — the varieties of most interest to mainstream algebra. What it buys is a commutator.
A commutator operation
In any congruence-modular variety there is a binary operation [θ, φ] on congruences generalising the group commutator, with the expected monotonicity and additivity properties.
Abelian, nilpotent, solvable
The commutator supports the full hierarchy, and an abelian algebra in a congruence-modular variety is polynomially equivalent to a module.
Krull–Schmidt
Uniqueness of direct decomposition into indecomposables holds for finite algebras in congruence-modular varieties.
The source's Chapter II §13 develops the centre of an algebra and points toward this theory. The general commutator for congruence-modular varieties was developed largely afterwards, by Freese, McKenzie, Gumm and Hagemann–Herrmann, and it is one of the principal developments the source's closing chapter anticipates.
Frequently asked questions
Does congruence-distributive imply congruence-modular?
Yes, since every distributive lattice is modular. The converse fails — groups are modular but not distributive.
Can a variety be congruence-distributive without being permutable?
Yes. Lattices are congruence-distributive and not congruence-permutable. The two conditions are independent, and having both is arithmeticity.
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 87-90.
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.