Boolean Constructions and Discriminator Varieties
Discriminator Varieties and their Structure
Varieties generated by classes of algebras sharing a discriminator term, and the representation theory that makes them, in the source's words, remarkably well-behaved.
Learning objectives
- Define discriminator variety
- State the Boolean product representation theorem
- Enumerate the properties discriminator varieties enjoy
The definition
A variety V = V(K) where K is a class of algebras admitting a common term t that acts as the ternary discriminator on every member of K.
The term must be uniform across K — a single term that specialises to the discriminator on each algebra. Individual algebras with discriminator terms are discriminator algebras; the variety notion requires uniformity.
| Variety | Discriminator source |
|---|---|
| Boolean algebras | 2 |
| V(A) for A primal | A |
| V(A) for A quasiprimal | A |
| Post algebras of order n | The generating Post algebra |
| Cylindric algebras of finite dimension | Standard construction |
| Monadic algebras | Standard construction |
| n-valued Łukasiewicz algebras | The generating chain |
The representation theorem
Every member of a discriminator variety is isomorphic to a Boolean product of simple algebras, each of which lies in ISPU(K).
Birkhoff's theorem gives a subdirect representation with subdirectly irreducible factors and no further control. Here the factors are simple, the index set is a Boolean space, and the representation satisfies gluing conditions. That is the maximum structural information a representation theorem can carry.
The properties enjoyed
| Property | Holds? |
|---|---|
| Congruence-permutable | Yes |
| Congruence-distributive | Yes |
| Arithmetical | Yes |
| Congruence extension property | Yes |
| Semisimple — all subdirect irreducibles are simple | Yes |
| Every member is a Boolean product of simple algebras | Yes |
| Amalgamation property | Yes, under mild hypotheses |
| Finitely generated case: decidable equational theory | Yes |
| Finitely generated case: finite equational basis | Yes |
Few classes of varieties satisfy all of these. The source's assessment — that probably no other class of varieties is so well-behaved yet so fascinating — reflects the combination of this list with the genuine variety of examples.
Why they matter beyond universal algebra
Algebraic logic
Cylindric algebras, monadic algebras and Post algebras are discriminator varieties. Their good behaviour is what makes algebraic treatments of quantifier logic tractable.
Many-valued logic
Łukasiewicz algebras of finite order are discriminator varieties, giving many-valued logics a complete algebraic semantics.
Decidability
Discriminator varieties sit on the decidable side of most decidability dichotomies, which Chapter V §5 explores.
Because every member decomposes into simple factors over a Boolean space, questions about a discriminator variety reduce to questions about its simple members plus Boolean algebra — and the theory of Boolean algebras is decidable. This is the mechanism behind the decidability results.
Frequently asked questions
Must the discriminator term be the same for all members of K?
Yes — uniformity is essential. A class where each algebra has some discriminator term, but no single term works throughout, does not generate a discriminator variety and the representation theorem fails.
Are all arithmetical varieties discriminator varieties?
No. Arithmeticity is strictly weaker. Discriminator varieties are arithmetical, but there are arithmetical varieties — including some varieties of Heyting algebras — with no discriminator term.
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.9, book pages 186-191.
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.