---
tag: 3M8Y76E4AG
kind: concept
name: Jónsson terms
msc: [08B05, 08B10]
sources:
  - url: https://en.wikipedia.org/wiki/Distributive_lattice
---

## Summary

Jónsson terms are a finite chain of ternary terms of a [[variety|AJNZJD8442]] that connect the first projection to the third by identities.
By Jónsson's theorem, a variety has Jónsson terms if and only if it is a [[congruence-distributive variety|9GZ7F6ETC6]].
They give a Mal'cev condition for congruence distributivity, so this property of all algebras of a variety reduces to the existence of finitely many terms.

## Explanations

### L1 human

Jónsson terms are a list of formulas with three inputs that step from the formula $x$ to the formula $z$.
Each formula returns $x$ when the first and the last input agree.
Neighbouring formulas agree when two particular inputs are equal.
For lattices, one middle step suffices: the majority formula, which returns the value that appears at least twice.