Boolean Constructions and Discriminator Varieties
The Primal Algebra Characterisation Theorem
The theorem identifying primal algebras by intrinsic conditions, and the representation of the generated variety by Boolean powers.
Learning objectives
- State the characterisation of primality
- Explain the role of the ternary discriminator
- Describe the Boolean power representation
The characterisation
A finite algebra A with at least two elements is primal if and only if it is simple, has no proper subalgebras, has no non-trivial automorphisms, and is quasiprimal — equivalently, if and only if the ternary discriminator is a term operation of A and the first three conditions hold.
Each condition removes an obstruction. Together they say the algebra has no internal structure that a term operation could be forced to respect.
| Condition | Obstruction if it fails |
|---|---|
| Simple | A non-trivial congruence must be preserved by every term operation, but not by every operation |
| No proper subalgebras | A subuniverse is closed under all term operations, but not under all operations |
| No non-trivial automorphisms | An automorphism commutes with every term operation, but not with every operation |
| Discriminator is a term | Supplies the case analysis needed to build arbitrary operations |
Congruences, subuniverses and automorphisms are the three basic kinds of relation preserved by term operations. Primality says there are none to preserve, so nothing constrains the term operations and they exhaust all operations.
The role of the discriminator
The operation t(x, y, z) equal to z when x = y, and to x otherwise.
The discriminator implements a conditional: it tests equality and branches. Given it as a term operation, arbitrary operations can be assembled by case analysis over the finitely many argument tuples.
It is the discriminator, not primality itself, that generalises. Dropping the conditions on subalgebras and automorphisms while keeping the discriminator gives quasiprimal algebras; requiring only that the discriminator be a term operation of a generating algebra gives discriminator varieties — the chapter's principal subject.
Boolean power representation
If A is primal, every algebra in V(A) is isomorphic to a Boolean power A[B]* for a unique Boolean algebra B, and this gives a categorical equivalence between V(A) and the variety of Boolean algebras.
So the whole variety is a copy of the variety of Boolean algebras. All questions about V(A) transfer to questions about Boolean algebras, where the answers are known.
| Question about <em>V</em>(<strong>A</strong>) | Answer |
|---|---|
| Subdirectly irreducibles | Only A |
| Subvarieties | Only the trivial variety and V(A) itself |
| Congruence lattice of a member | The filter lattice of the corresponding Boolean algebra |
| Equational theory | Decidable — check in A |
| Finite basis | Exists |
| Free algebra on n generators | A[free Boolean algebra]* |
Historical significance
Foster's work on primal algebras in the 1950s was the origin of the whole Boolean-methods programme in universal algebra. The observation that a single algebra could generate a variety equivalent to Boolean algebras suggested that Boolean techniques might apply far more widely.
The successive generalisations — quasiprimal algebras (Pixley), Boolean products, discriminator varieties — each widen the class of varieties amenable to these techniques while preserving as much of the structure theory as possible. The source describes discriminator varieties as remarkably well-behaved and unusually interesting, and this lineage is why.
Frequently asked questions
Is the Boolean algebra B in the representation unique?
Yes, up to isomorphism. It is recovered as the Boolean algebra of factor congruences of the given algebra.
Does the characterisation give an effective test for primality?
For a given finite algebra, yes — simplicity, subalgebras and automorphisms are all finite checks, and whether the discriminator is a term operation is decidable by generating the clone.
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.7, book pages 169-173.
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.