---
tag: MC9MX40Z7S
kind: concept
name: Discriminator variety
msc: [08B05, 08B26]
specializes: [GW78A05TNA]
sources:
  - book: isbn:9780387905785
---

## Summary

A discriminator variety is a [[variety|AJNZJD8442]] generated by algebras that share one term acting as the [[ternary discriminator|72PG6XCQ7D]] on each of them.
Boolean algebras, the varieties generated by a finite field, and the varieties generated by a quasi-primal algebra are examples.
Discriminator varieties are arithmetical, and each of their subdirectly irreducible members is simple.
Their algebras can be studied as Boolean products of simple algebras, much as Boolean algebras are studied through Stone spaces.

## Explanations

### L1 human

In some algebras one can write, with the given operations, a formula that tests whether two elements are equal.
A discriminator variety is built from algebras that all use the same formula for this test.
The two-element Boolean algebra is the basic example: the test is "if $x$ and $y$ agree, output $z$, else output $x$", and Boolean operations can express it.
Having such a test makes the variety behave much like the variety of Boolean algebras.