Boolean Constructions and Discriminator Varieties
Primal Algebras and Functional Completeness
Finite algebras whose term operations are all possible operations, and the remarkable rigidity this forces.
Learning objectives
- Define primal algebra and verify the two-element example
- State the consequences of primality for the generated variety
- Distinguish primality from related conditions
Definition
A finite algebra A with at least two elements such that every finitary operation on A — every function An → A for every n — is a term operation of A.
Equivalently: the clone of A is the full clone of all operations on A. Nothing is missing.
The motivating example
2, the two-element Boolean algebra, is primal. Every Boolean function of n variables is expressible in disjunctive normal form using ∨, ∧ and ′ — so every operation on {0, 1} is a term operation.
This is the algebraic content of the functional completeness of the standard connectives in propositional logic.
| Algebra | Primal? | Reason |
|---|---|---|
| 2, Boolean | Yes | Disjunctive normal form |
| A finite field Fq as a ring | No | Ring terms give only polynomial functions; not all functions are polynomial in more than one variable without additional operations |
| Fq with all polynomial operations named | Yes | By Lagrange interpolation |
| Z/4 as a ring | No | Term operations are polynomials with integer coefficients |
| Post algebras of order n | Yes | By construction |
| Any finite lattice with 3+ elements | No | Lattice terms are monotone; non-monotone operations are unreachable |
Lattice operations are order-preserving, so every lattice term operation is monotone. Since most operations are not monotone, no lattice with more than two elements can be primal. Obstructions of this kind — preserved relations — are exactly what the characterisation theorem formalises.
Consequences of primality
If A is primal, then V(A) has the same structure as the variety of Boolean algebras: it is arithmetical, semisimple, congruence-distributive, congruence-permutable, and A is its unique subdirectly irreducible member.
Every member of V(A) is isomorphic to a Boolean power A[B]* for some Boolean algebra B. So the variety is completely classified: its members correspond bijectively to Boolean algebras.
Primality is the strongest possible finiteness condition on a finite algebra, and it delivers a total classification of the generated variety. This is the template that discriminator varieties generalise.
Frequently asked questions
Are there primal algebras of every finite size?
Yes. For each n ≥ 2 the Post algebra of order n is primal, so primal algebras exist at every finite cardinality above one.
Can an infinite algebra be primal?
The definition requires finiteness. For infinite algebras there are more operations than terms by a cardinality argument, so the condition is unsatisfiable as stated.
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 169-172.
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.