Boolean Constructions and Discriminator Varieties
Arithmetical Varieties and Pixley Terms
Varieties that are both congruence-permutable and congruence-distributive, and the single term that characterises the combination.
Learning objectives
- Define arithmetical variety
- State Pixley's term characterisation
- Identify arithmetical varieties and their properties
The definition
A variety that is both congruence-permutable and congruence-distributive.
The name reflects the fact that the congruence lattices behave like the lattice of ideals of a principal ideal domain — distributive, with permuting elements — which is the situation in elementary arithmetic.
A variety is arithmetical if and only if there is a ternary term p satisfying p(x, y, y) ≈ x, p(x, y, x) ≈ x, and p(x, x, y) ≈ y.
A ternary term satisfying those three identities.
The first and third identities are exactly the Mal'cev conditions for permutability. The middle identity — p(x, y, x) ≈ x — is what adds distributivity. So a Pixley term is a Mal'cev term with one extra requirement.
Relation to the discriminator
The ternary discriminator satisfies the second and third Pixley identities but not the first: t(x, y, y) equals x when x ≠ y, and equals y when x = y — which is x in both cases. So the discriminator is a Pixley term.
Since the discriminator is a Pixley term, every discriminator variety is arithmetical. The converse fails: arithmeticity is strictly weaker.
| Condition | Term requirement | Strength |
|---|---|---|
| Congruence-permutable | Mal'cev term | Weakest |
| Arithmetical | Pixley term | Middle |
| Discriminator variety | A term acting as the discriminator on generators | Strongest |
Examples
| Variety | Arithmetical? | Discriminator variety? |
|---|---|---|
| Boolean algebras | Yes | Yes |
| Heyting algebras | Yes | No |
| Post algebras | Yes | Yes |
| Cylindric algebras of finite dimension | Yes | Yes |
| Groups | No — not congruence-distributive | No |
| Lattices | No — not congruence-permutable | No |
| V(A) for quasiprimal A | Yes | Yes |
Heyting algebras are arithmetical but not a discriminator variety. They are the standard witness that the implication does not reverse, and their structure theory is correspondingly less complete than that of Boolean algebras.
Consequences of arithmeticity
- The Chinese remainder theorem holds. For congruences θ1,…,θn and compatible elements, there is a common solution. This is where the name comes from.
- Congruence lattices are distributive and permuting, so joins are relational products and the lattice structure is as simple as it can be.
- Jónsson's lemma applies, since arithmetical implies congruence-distributive.
- Directly indecomposable factors are well behaved, and the Boolean algebra of factor congruences is a complemented sublattice of the congruence lattice.
The classical Chinese remainder theorem for integers relies on coprime moduli — distributivity — and on solvability of simultaneous congruences — permutability. Arithmetical varieties are exactly those where both ingredients are present, which is why the classical statement generalises verbatim.
Frequently asked questions
Is every congruence-permutable and congruence-distributive variety arithmetical?
Yes — that is the definition. The content of Pixley's theorem is that the conjunction is captured by a single term rather than requiring both Mal'cev conditions separately.
Are there arithmetical varieties that are not finitely generated?
Yes. Heyting algebras form an arithmetical variety that is not generated by any finite set of finite algebras.
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.10, book pages 196-199.
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.