---
tag: BV2GCXPDDJ
kind: definition
name: Discriminator variety
msc: [08B05, 08B26]
defines: MC9MX40Z7S
uses: [AJNZJD8442, 72PG6XCQ7D, N9M2N27END, GW78A05TNA]
sources:
  - book: isbn:9780387905785
---

## Definition

Let $\mathcal{K}$ be a class of algebras of the same signature.
Assume that there is a ternary [[term|N9M2N27END]] $s(x, y, z)$ whose term operation on every $A \in \mathcal{K}$ is the [[ternary discriminator|72PG6XCQ7D]] on $A$.
The [[variety|AJNZJD8442]] $V(\mathcal{K})$ generated by $\mathcal{K}$ is then called a *discriminator variety*, with discriminator term $s$.
A variety is a discriminator variety if it is of the form $V(\mathcal{K})$ for such a class $\mathcal{K}$.

## Remarks

Example: the variety of Boolean algebras is generated by the two-element Boolean algebra $\mathbf{2}$.
On $\mathbf{2}$, the term $s(x, y, z) = (x \wedge (x \oplus y)) \vee (z \wedge \neg(x \oplus y))$, with $x \oplus y = (x \wedge \neg y) \vee (\neg x \wedge y)$, is the discriminator.
Check: if $x = y$ then $x \oplus y = 0$ and $s = z$; if $x \neq y$ then $x \oplus y = 1$ and $s = x$.
Non-example: the variety of lattices is not a discriminator variety.
A discriminator term $s$ satisfies $s(x, y, y) \approx x$ and $s(x, x, y) \approx y$ on $\mathcal{K}$, hence on $V(\mathcal{K})$.
So a discriminator variety is congruence-permutable, while the three-element chain is a lattice with non-permuting congruences.
The identities $s(x, y, y) \approx x \approx s(x, y, x)$ and $s(x, x, y) \approx y$ show that $s$ is a Pixley term, so every discriminator variety is an [[arithmetical variety|GW78A05TNA]].
The term $s$ need not act as the discriminator on every member of $V(\mathcal{K})$; for Boolean algebras it does so only on $\mathbf{2}$ and the trivial algebra.