---
tag: QWSBDFM47K
kind: concept
name: Quasi-primal algebra
msc: [08A40, 08B05]
sources:
  - url: https://en.wikipedia.org/wiki/Functional_completeness
---

## Summary

A quasi-primal algebra is a finite algebra in which the [[ternary discriminator|72PG6XCQ7D]] is a term operation.
Every primal algebra is quasi-primal, and so is every finite field.
The variety generated by a quasi-primal algebra is a [[discriminator variety|MC9MX40Z7S]], so it is arithmetical and well understood.
Quasi-primal algebras are a natural middle ground between primal algebras and arbitrary finite algebras.

## Explanations

### L1 human

In a quasi-primal algebra, the operations can test whether two elements are equal and act on the answer.
This is weaker than expressing every function, as in a primal algebra.
For example, the field with four elements can express the test, but it cannot express a constant that is moved by its Frobenius automorphism.
Such algebras still behave much like Boolean algebras in many structural questions.