---
tag: XP67WCPQ59
kind: concept
name: Polynomial operation
msc: [08A40]
sources:
  - url: https://en.wikipedia.org/wiki/Clone_(algebra)
---

## Summary

A polynomial operation of an algebra is a [[term operation|N9M2N27END]] in which some arguments are fixed to constant elements of the algebra.
For a commutative ring, the polynomial operations are the [[polynomial functions|1G0B73WPX8]] in several variables, which explains the name.
Polynomial operations describe the congruences of an algebra: an equivalence relation is a congruence exactly when it is preserved by all unary polynomial operations.

## Explanations

### L1 human

Term operations may use only the operations that the algebra comes with.
Polynomial operations may also use any element of the algebra as a fixed constant.
In the group $(\Z, +)$, the map $x \mapsto 2x$ is a term operation, while $x \mapsto 2x + 5$ is a polynomial operation that is not a term operation.
In a ring, $x \mapsto 3x^2 + 7$ is a polynomial operation, matching the usual polynomial.