---
tag: XD01Y7B6XW
kind: concept
name: Primal algebra
msc: [08A40]
specializes: [QWSBDFM47K]
sources:
  - url: https://en.wikipedia.org/wiki/Functional_completeness
---

## Summary

A primal algebra is a finite algebra in which every finitary operation on its universe is a term operation.
Equivalently, its basic operations form a [[functionally complete|TDZVCWYJ9F]] set.
The two-element Boolean algebra and the rings $\Z_p$ for a prime $p$ are primal.
By [[Foster's theorem|TCD8SZW95Z]], the variety generated by a primal algebra looks just like the variety of Boolean algebras.

## Explanations

### L1 human

In a primal algebra, the given operations can express every function on the set.
The two-element Boolean algebra is the model: every truth table is a Boolean formula.
Another example is arithmetic modulo a prime $p$: every function on $\set{0, 1, \dots, p - 1}$ is a polynomial with integer coefficients, reduced modulo $p$.
For instance, modulo $3$ the function equal to $1$ at $0$ and $0$ elsewhere is $1 - x^2$.