Boolean Constructions and Discriminator Varieties
The Ternary Discriminator Function
The operation that tests equality and branches, and the reason it is the single most consequential term operation in the subject.
Learning objectives
- Define the ternary discriminator and compute with it
- Show how it expresses case analysis
- Explain why its presence forces strong structural properties
The definition
On a set A, the operation t: A3 → A given by t(x, y, z) = z if x = y, and t(x, y, z) = x otherwise.
Read as a conditional: if x equals y then return z else return x. It is the algebraic form of a branching instruction.
| Case | <em>t</em>(<em>x</em>, <em>y</em>, <em>z</em>) |
|---|---|
| x = y | z |
| x ≠ y | x |
| t(x, x, z) | z |
| t(x, y, x) | x — always |
| t(x, y, y) | y if x = y, else x |
Immediate consequences
An algebra with the ternary discriminator as a term operation generates an arithmetical variety — both congruence-permutable and congruence-distributive.
Arithmeticity normally requires verifying two independent Mal'cev conditions. The discriminator delivers both at once, which is the source of its power.
Simplicity
A non-trivial algebra having the ternary discriminator as a term operation is simple.
Suppose θ is a congruence with a θ b for some a ≠ b. For any c, d, apply the substitution property to the discriminator: t(a, b, c) = a since a ≠ b, while t(a, a, c) = c. Since a θ b, these are θ-related, so a θ c for every c — hence θ = ∇.
The proof is a template. Having the discriminator means one can always write “if these two elements are equal do this, otherwise do that” as a term, and then apply the substitution property to both branches. Almost every argument about discriminator varieties uses this move.
Where the discriminator occurs
| Algebra | Discriminator is a term? |
|---|---|
| 2, Boolean | Yes |
| Any primal algebra | Yes — every operation is a term |
| Any quasiprimal algebra | Yes — by definition |
| A finite field with all polynomial operations | Yes |
| Cylindric algebras of finite dimension | Yes |
| A finite simple group | No — group terms cannot test equality |
| Any lattice with 3+ elements | No — the discriminator is not monotone |
The discriminator is not order-preserving, so no algebra whose term operations are all monotone can have it. This immediately rules out lattices, semilattices and ordered structures generally.
Frequently asked questions
Is the discriminator a single operation or a family?
A single ternary operation, determined by the underlying set. Any set of size at least two has exactly one ternary discriminator.
Can an infinite algebra have a discriminator term?
Yes. The definition does not require finiteness, and infinite discriminator algebras exist — though the strongest structure theorems are usually stated for the finite case.
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 186-188.
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.