---
tag: 5C91SFVTY7
kind: concept
name: Pixley term
msc: [08B05, 08B10]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence-permutable_algebra
---

## Summary

A Pixley term of a [[variety|AJNZJD8442]] is a ternary term $p$ such that $p(x, y, y) = x$, $p(x, x, y) = y$ and $p(x, y, x) = x$ hold.
By Pixley's theorem, a variety has a Pixley term if and only if it is an [[arithmetical variety|GW78A05TNA]].
A Pixley term is a [[Mal'cev term|Y203PPTS61]] that also yields a [[majority term|9879ZMW5AA]].

## Explanations

### L1 human

A Pixley term is a three-input formula that behaves like the Mal'cev term of groups, and in addition returns $x$ on inputs $x, y, x$.
For Boolean algebras it can be written with "and", "or" and "not".
Its existence makes the congruences of every algebra in the variety both permute and satisfy the distributive law.