Boolean Constructions and Discriminator Varieties

Primal Algebras

A primal algebra has every possible operation as a term operation. It generates a variety categorically equivalent to Boolean algebras — the strongest possible statement that an algebra behaves like the two-element Boolean algebra.

Engineering · Mathematics4 min readKV-MATH-0239

Learning objectives

Define primality and verify it for the two-element Boolean algebra.
State Foster's theorem on the variety generated by a primal algebra.
Show every member of that variety is a Boolean power of the generator.
Distinguish primal from quasiprimal and from functionally complete.
Explain how primality is checked in practice.
Recognise why primality is rare.

01Primality

A finite algebra A with at least two elements is primal when every finitary operation on its universe is a term operation.

A primal ⟺ for every n and every g : Aⁿ → A, g = p^A for some term p

The clone of term operations is the full clone of all operations on A. Nothing is missing.

Key resultThe 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 of 2. This is the archetypal example, and it explains why primality is the right generalisation of 'behaves like 2'.

Primality is an extremely strong condition. It says the algebra's term operations exhaust every conceivable function on its universe, so no structure is left unexpressed. Almost no naturally occurring algebra is primal.

02Foster's theorem

The variety generated by a primal algebra is as well understood as a variety can be.

Key resultFoster's theorem

If A is primal, then V(A) is categorically equivalent to the variety of Boolean algebras, and every member of V(A) is isomorphic to a Boolean power A[B]* for a unique Boolean algebra B.

A is the unique subdirectly irreducible
Primality forces the only subdirectly irreducible member of V(A) to be A itself, mirroring 2 in the Boolean case.
Every member is a subdirect power of A
By Birkhoff's subdirect representation theorem, combined with the previous point.
The subdirect power is a Boolean power
Primality supplies enough term operations to force local constancy, upgrading the subdirect representation to a Boolean power.
The Boolean algebra is unique
So V(A) and Boolean algebras correspond, member by member, with matching homomorphisms.

03What categorical equivalence buys

Transferred properties
Property of Boolean algebras	Holds in V(A) for A primal
One subdirectly irreducible	Yes — A itself
Arithmetical	Yes
Congruence-distributive	Yes
Congruence-permutable	Yes
Congruence extension property	Yes
Locally finite	Yes
Finitely based	Yes
Decidable first-order theory	Yes
Every member a Boolean power of the SI	Yes
Lattice of subvarieties: two elements	Yes — trivial and the whole variety

Every structural question about V(A) reduces to the corresponding question about Boolean algebras, which is settled. This is the strongest form of the transfer principle the chapter is built around.

04Checking primality

ProcedureDeciding whether a finite algebra is primal

in: finite algebra A → out: primal or not

input: finite algebra A, |A| = n ≥ 2
necessary conditions, cheap to check first:
A is simple (else a congruence blocks some operation)
A has no proper subalgebras (else a subuniverse blocks some operation)
A is rigid: the only automorphism is the identity
sufficient test (Rosenberg): A is primal iff its clone is contained in
no maximal clone — check against the finitely many maximal clones
practical route: verify the discriminator is a term operation, plus
that A is simple, subalgebra-free and rigid
output: primality decision

Rosenberg's classification of maximal clones on a finite set makes the test finite and effective. Caveat: the number of maximal clones grows rapidly with |A|, so the test is expensive for larger algebras; UACalc implements the practical version.

CautionThe three necessary conditions are not obviously sufficient

Simplicity, no proper subalgebras and rigidity are all necessary for primality, and it is tempting to assume they suffice. They do not on their own — an algebra can satisfy all three and still have a clone missing some operations. The discriminator condition is what closes the gap, and that is the content of the next pages.

05Why the three conditions are necessary

Simplicity

Congruences block operations

If θ is a proper non-trivial congruence, every term operation respects it. An operation violating θ cannot be a term operation, so A is not primal.

No proper subalgebras

Subuniverses block operations

If C is a proper subuniverse, every term operation maps C into C. A constant operation with value outside C, restricted to C, violates this.

Rigidity is the same argument applied to automorphisms: every term operation commutes with every automorphism, so a non-trivial automorphism excludes any operation that fails to commute with it. Together the three say A has no internal structure of any kind for term operations to respect — which is exactly what primality demands.

06Primal, quasiprimal, functionally complete

Three related conditions, in decreasing strength, each generalising a different aspect of primality.

The hierarchy
Condition	Requirement	Generated variety
Primal	every operation is a term operation	categorically equivalent to Boolean algebras
Quasiprimal	the discriminator is a term operation	a discriminator variety
Functionally complete	every operation is a polynomial operation	no such clean characterisation

Quasiprimality relaxes primality by requiring only that one particular operation — the ternary discriminator — be a term operation. Functional completeness relaxes it by allowing constants. The two relaxations go in different directions, and the following pages treat each in turn.

Frequently asked

Are there primal algebras other than 2?

Yes, for every finite size. The n-element algebra with all operations taken as basic is trivially primal, and less artificial examples exist — Post algebras and certain finite fields with extra operations are primal. What is rare is for a naturally arising algebra with a small signature to be primal.

Is a finite field primal?

Not as a field in the ring signature — the field operations do not generate every operation, since term operations of a finite field are polynomial functions and not every function is one for fields larger than the prime field. Adding the right extra operations makes it primal, and Arens and Kaplansky's work on filtered Boolean powers of finite fields is the classical study of the near-miss.

Does primality imply the variety is finitely based?

Yes, via the categorical equivalence with Boolean algebras, which are finitely based. It also follows independently from Baker's theorem, since the variety is congruence-distributive and generated by a finite algebra.

Sources and further reading

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.

Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.